The 10-megaton nuclear device, in the most probable outcome, would have caused serious but not irreparable damage to the Mother Ship. The world-ship was, after all, some 550 kilometers wide.  Unfortunately, due to an unforeseen design error, the main Small Craft Embarkation Area was only about 8 kilometers from the space-twisting drive, which was destabilized by the blast.
As a result, the ship's mass was accelerated rather violently in all directions at once: the majority of those directions being space-time other than our own, which, combined with the fact that what mass was in our dimension was usually moving beyond earth's escape velocity, meant that although the giant ship was in a near-earth geosynchronous orbit a mere 22,000 or so miles up only a small proportion of its mass actually hit the earth (and that was generally in pretty small pieces).
Still, it was an awfully large ship.
It is the year 2020: fifteen  years have gone by. The world is still a mess. Some 250 million people were killed in the initial invasion, rather more than 100 major cities were destroyed: fifteen million more died from chunks of the Mother Ship hitting the earth, and close to a billion died of famine and cold and disease as a result of all the crap knocked into the atmosphere, mostly in poorer countries. Nearly a billion more were at least temporarily displaced.
The "world government" remains somewhat shaky, and the proposed "united states of Earth" has not yet been able to get off the ground. In part this is due to somewhat ambiguous attitudes towards the US, which may have been home to the guys who blew up the Mother Ship and ended the invasion, but also carefully hid the alien technology that could have stopped the invasion for half a century from the rest of the world (not to mention from the vast majority of its own scientists), and was also home to the guys that blew up the Mother Ship, thereby raining down spaceship rubble which led to the horrors of the Three Cold Years (not a full blown nuclear winter, but three years of abnormally cold growing seasons really is Not Good: the first year was the worst, since there was little time to make preparations).
Not to mention the fact that the US has refused to explain how it managed to get a nuke aboard the Mother Ship: paranoid explanations are numerous. This is in fact not the fault of the US: David Levinson never was able to explain afterwards how he had been able to hack an entirely alien operating system overnight, and the data (both on his laptop and on the copies made for the government) upon re-examination turned out to be entirely illegible. His father ascribed it to a perfectly ordinary miracle: others were less amiable, and only thanks to direct presidential intervention was David able to avoid Enhanced Interrogation.
(The Muslims claim that the US is making it up, and they're not the only ones, although the Muslims are the only ones that claim Allah did it. The survival of Mecca rather low on the alien's target list just confirms this to many).
In a dimension just a little bit over the roof of our universe, Proto-Sapient Monitor Xz(blue wavelength of light) is a bit miffed over the reprimand it received due to the "destructive" and "sloppy" methods employed to prevent the extinction of its charges: after all, Semi-Sapient Species 16752369 code name "Locusts" is low-priority for conservation, and it's not like a dozen other hives aren't roaming this galaxy.
The US was hard hit, but was able to establish a functional emergency government and rationing system, and people nationwide organized themselves with surprising skill in the absence of representatives of the federal government. Short on food itself due to the abnormally cold weather and the breakdown of various links in the advanced infrastructure on which industrial farming depends, the US was unable to help much through the Cold Years, but did provide a lot of technological support and scientific counseling for other nations struggling to find ways to feed their populations. (The US made a colossal greenhouse-building effort when it became clear the shitty weather was going to last, and made some remarkable steps in extracting nutrition from everything from wood pulp to coal, which they then helped others duplicate).
Although there were a few communities and states which went a little batshit in the aftermath, by 2020 the US is back on its feet. People complain about how goddamn cold the new capital of Minneapolis-St. Paul is as they used to complain about how hot Washington used to be, and many of the ruined or partly ruined cities are being slowly rebuilt: the alien weapons left no radioactivity (a fact that annoys Houston, Texas survivors). Due to the majority of the deaths being from urbanites, it's a more conservative country, although fortunately a less nutty Republican party is not trying to take advantage of this to reinstate the Gilded Age. In any event, the importance of people working together and "socialistic" measures during the Cold Years impressed itself clearly enough that nobody thinks government needs to be drowned in a bathtub.
The Middle East had a
varied experience. Egypt was largely wiped out (after blowing up Cairo, one of the aliens had a brainwave and blew up the Aswan dam), while Iran, in spite of the loss of its two biggest cities and massive famine during the Cold Years survived and remains under an Islamicist regime (conservative country folk being overrepresented among the survivors). The Tel Aviv attack killed nearly a third of the population of Israel:  however, the alien invasion, at least for a while, united the Jews and the Arabs, as did the extension of Israel's heroic efforts to produce enough food to its Palestinian subjects. The new Israeli-Palestinian Confederation is actually a functional state, although the status of the bi-national city of Jerusalem still occasionally excites sore feelings.
There was no 9/11 in this TL, and Saddam Hussein was still in Baghdad when it blew up: civil war followed, followed by famine, followed by more war. There are currently three Sunni and two Shi'a successor states, the Kurdish Sunni one being under Turkish military occupation. The rather fragile Syrian and Lebanese states fell apart: the Turks and the Confederation are now working together to keep peace in the area. Egypt lost 80% of its population and is now run by the Muslim Brotherhood, but at least the now returned Nile floods bring a rich layer of soil every year, making for good harvests.
Saudi Arabia, neither a center of technological innovation nor a nation particularly favorable to farmers, suffered very badly with the global food crisis - although there were still those willing to buy oil, there were few with food to sell: rationing soon set in, and after the civil war broke out and supply networks broke down, the loss of life was rather impressive. The Arabian Federation is a great-power sponsored work in progress since 2009, consisting of the Hejaz, Nejd, Nefud and El-Hasa autonomous regions: the 2/3 decline in population has at least for the nonce taken the starch out of the more fanatical sects. (Yemen, which suffered at least as badly, still has the energy to fiercely dispute their northern border with the Hejazis).
Europe had a harder time than the US, with less in the way of a coordinated response and a more concentrated (and therefore more blow-up-able) population: quite a bit of the continent, mostly in Eastern Europe, descended into chaotic conditions in the first year after the invasion, and even in the west the need to deal with millions of refugees whose homes were ash led to a series of social and political crises, most notably the neo-fascist takeover in Italy and the bloody French Camp Riots (the immigrant population, with some justification, felt that white native-born French were getting all the new housing and the best of the food). Still, human and technological resources were such that western Europe managed to feed itself, if a bit marginally at first: Western European integration had taken a big step forward by the end of the Cold Years, although Eastern Europeans remain rather resentful that the westerners did rather little to help them directly (although, like the US, they did provide technical aid) in the initial period of the Cold. British unity has been somewhat enhanced by the move of the government to Edinburgh.
China is still run by the Communist Party, albeit with the Army greatly strengthened as an independent power center: the Communist Party Heads, not giving two farts about public opinion, got the hell out of Beijing as soon as they noticed the Giant Alien Spaceship hovering overhead. And when the famines began, it was the Army and Party which instituted and maintained the draconian regulations and controls which managed what food supply was available and mobilized the mass forces required for such efforts as the chemical processing of every tree and shrub in China into more-or-less edible goo. ("Triage" of disloyal or unproductive provinces took place during the Cold Years: the survival rates of Tibetans, for instance, were a lot lower than those among Han Chinese). Violence and social breakdown took place in some provinces, but by 2010 the central government was once more firmly in charge. China remains rather impoverished, though: trade took nearly a decade to fully revive, in spite of international efforts to promote it.
India, with a less brutal and engineer-filled government than China, and almost as much damage to its population centers, was more half-assed in its response to the disaster, and the most effective responses were generally local efforts. Famines were very bad, especially in the north, and the country descended into chaos with local islands of stability: although India has reemerged in the last decade as a united country, it is a far looser and more federal state, state borders have changed, and southern India, which now makes up an enlarged portion of the population, is more influential. The depopulated wreck of Nepal has been absorbed into India, while Pakistan broke up entirely (10 million armed Afghan refugees did not help).
Russia survived, although the usual Russian lack of organization meant that the Stalinist methods of rationing, food production, and resource management were not really carried out with much skill beyond the remaining sizeable urban centers. As a result, much of the country, especially the far north and Siberia, became heavily depopulated and sank into chaos: the nation is nowadays increasingly dominated by the Children of Jesus, a religious movement emphasizing aid to others, the brotherhood/sisterhood of all Russians, and the importance of children (large numbers of children).
The Japanese took proportionally very high losses in the war, due to the concentration of their population, and suffered a massive hit to their infrastructure and industry: on the other hand, order was maintained even during the worst of the Cold Years (in which Japanese fishing ships essentially removed every gram of protein from the North Pacific and converted every apartment into a mini-greenhouse and even raised insects for food), during which only a few Japanese actually starved to death while everyone _almost_ starved (Pretty much everyone born during those years suffers from developmental damage due to malnutrition). In the postlude, a kind of militance has arisen in younger Japan, as the young who lived through the period have come to rebel against what they see as the conformity and obedience of their elders in the face of catastrophe (their parents grumble that the kids don't know what they are talking about it was self-discipline and selflessness by which Japan survived!)
Latin America, although less dependent on food imports than other parts of the world (especially since they weren't exporting much of anything) and often able to expand their land under cultivation, still suffered from serious food shortages, which combined with the destruction of its often oversized and over-centralized capitals, led to political chaos and breakdown through much of the area. After the first few months Mexico received increasing assistance from the US (desperate to avoid an even worse refugee crisis if Mexico were to collapse entirely and send tens of millions fleeing for supposed better conditions to the north), and although millions still _did_ flee, the situation was at least partially contained: like the western EC, NAFTA emerged considerably strengthened from the trial. In parts of Central America and Andean America, the central state pretty much ceased to exist, leading to great famines and over a dozen revolutions: a few odd 'socialistic' states still survive, and the two Brazils only reunited three years ago.
Africa south of the Arab lands is, as predictable, a mess, although the major nations, working through the World Government, have been doing their best to alleviate the suffering and reestablish working governments.
Said World Government, located in Switzerland (no big cities was advantageous), is something of a global NATO (lest the aliens return), a global trading organization, a global development program far more muscular than the rather feeble pre-invasion organizations, and, supposedly, the precursor of a true World Government. However, the format of such a World Government remains elusive, and given international suspicions and hostilities and tensions between rich and poor and democratic and authoritarian, etc., it is hard to see how one can be agreed on: a more extended war might have developed a practice of international community, but since it was all over in a few days, asides from the Numb Peace in the Middle East and the union of the Koreas arising from the failed invasion by a starving-to-death north, no fundamental increase in international amity has arisen. There is a much greater consciousness of our common humanity, but, as they say, that doesn't change the price of (in many cases fished to extinction) fish.
There has been a major increase in international openness and communication when it comes to science and technology: there are (at least in theory) no secret research centers, and there is energetic international research into every aspect of the alien's technology: so far, the fear the aliens or something like them might return has overridden national self-interest. (The Chinese, admittedly, haven't mentioned what they dug up in inner Mongolia in 1969, but then they never managed to put it back together again, and it would be kinda embarrassing to mention it at this point).
Not all of the aliens perished.
Some destroyer ships did receive messages conveying the alien equivalent of "F---, one of them just flew into the zap gun it's blowing uuup
" before this strategy could be employed, and the ships were just too goddamn large for conventional weapons to do more than superficial damage to (although the less environmentally conscious Soviets and Chinese simply nuked the hell out of them the minute the force-fields went down). Of course, all surviving destroyer ships simply crashed to earth once the sub-space power flow from the Mother Ship was cut off , and the shock of a mile drop is rather lethal even if the outer hull of your ship is strong enough to stay largely in one piece.
Unlike the destroyer ships, whose power consumption is always vast (you try holding a 15 mile wide ship aloft and immobile), the attack craft have some reserve power. Two cigar-smoking heroes managed to get back to Earth before crashing, and a number of alien pilots escaped into the safety of space before running out of power. Then it was just a matter of resetting their receivers to the Moon Base frequency
The Moon Base, carefully concealed beneath the surface of the dark side, had been established back in the 1940s as a forward base to study the Earth while the Mother Ship finished pillaging the resources of a nearby solar system, and was the launching point of the flying saucers which pestered the Earth in subsequent decades. (The Mother Ship on arriving had done a flyby of the Moon and the base as a morale-boosting thing for those brave hermaphrodites and neuters in the Planet Scouts).
The Moon base had its own broadcast power system for the exploratory saucers, operating on its own sub-space frequencies (which were different by 1982, when the US finally managed to cobble together the crashed ship, than when it had landed in 1947: if the Area 51 investigators had just known how to switch the receiver control to "search" for a few minutes, they could have had a working ship during the Reagan administration). There was a nasty bit when the power went off from the Mother Ship and they had to cold-start their own generator, but aside from the aliens in the anti-gravity elevator, there were no deaths.
On the part of the Moon base, there have been no efforts to rescue aliens stranded on Earth: secrecy is currently the watchword. There have been signals sent out in hopes of contacting another world-ship, but there may be no response for years or even decades. In the meantime, the aliens continue to spy on Earth, and draw plans for the extermination of humanity
Some skilled alien pilots with ships too damaged to escape managed to glide their ships into "soft" crashes on vegetation or in shallow water: some managed to hide for lengthy periods of time before being killed or captured. Those few aliens captured alive have not proven to be particularly useful sources of information, generally preferring a "death before dishonor" approach to interrogation, and in event tending to die rather quickly due to their inability to digest terrestrial foodstuffs not processed by a Mother Ships bio-reactors. Only four of them remain alive, suffering from various nasty nutritional diseases as a result of only partially successful efforts to synthesize alien-fodder from the alien's own biochemistry and the equivalents of MRE's taken from crashed ships (comparatively little food from the destroyer ships survived the crashes, subsequent explosions, and the rather long periods without refrigeration before human scientists found and identified them).
They interact with humans by way of robotic waldoes and cameras: no human risks coming within the effective range of their telepathic control (which, admittedly, is a bit spotty: those under alien mind-control tend to be a twitchy, convulsive lot), and are prodded and shocked by robot limbs to move to separate sealed quarters when it is time for clean-up and maintenance. All have been poor subjects for technological interviews, claiming pig-ignorance of anything but flying combat ships: torture and extended interrogation have been somewhat hampered by their apparent ability to place themselves in a coma-like state at will, not to mention ignorance of what will hurt an alien and what will kill an alien. Only slowly and through a great deal of careful annoyance, privileges given and withdrawn, extended tedium and endless conversation,  have bits and pieces of alien technological and scientific know-how emerged. 
After 15 years, reverse-engineering of alien tech has been slow, in spite of those clues winnowed from the aliens and the entire brainpower of the planet rather than a small subset of US scientists working on it. The major problem of broadcast power was finally licked in 2015, some years after a small "field" power broadcaster was found relatively intact abroad one of the crashed ships: although huge and crude, the reverse-engineered generator can now be used to activate different kinds of alien machines for experimental purposes (making earth tech receive and use the power has been more of a challenge). And although alien weapons and matter-energy converters remain unduplicated, a 2000-ton mass of machinery receiving the entire power output of a large nuclear reactor through a cable as thick as a man's thigh achieved negative weight for 0.5 seconds on May 11, 2020. It's all just a matter of refinement from here on, the jubilant engineers proclaim.
In other developments, analysis of an alien sandwich wrapper has led to a revolution in the synthetics industry.
 Not, however, ¼ the mass of the moon which is, after, 3476 kilometers wide and quite solid the initial mass estimates were bollixed more than a factor of magnitude by the fact that the effects of its propulsion system look a lot like a strong gravitational field from a distance.
 As in the "ID Aftermath" thread, I agree it's rather unlikely that a Gulf War vet would make his way to the Presidency only 5 years after the war.
 Israel's population is heavily concentrated in the Tel Aviv Metro area.
 Much invader equipment runs off sub-spatial energy fields generated by big-ass sources as much as several million miles away. As such, it is very hard to experiment on, since it is powered by "receiver nodes" actually part of the internal structure of the equipment: there are no wires, and no way to install them except by taking the machine apart, often in ways it is not supposed to be (which allows you to put current through various bits and see how they react, but not run an assembled machine).
 Carried out by keyboard, since the aliens have no vocal chords and nobody wants to play ventriloquists dummy to an alien telepath. Forcing the aliens to learn written English/Russian/whatever took years, and tricking them into admitting they understood it took even longer.
 The aliens, in the meantime, have learned a hell of a lot of new information about humans.