A nuclear war involving hundreds of detonations would most likely be a global war in which countries would be hard pressed to get involved rendering neutrality virtually impossible. In such an event I would expect that human civilization as a whole would suffer such a serious blow that its own extinction might happen it the medium to long term.
The first wave of nuclear detonations would likely target military and industrial infrastructures, as each side of the conflict would try to limit the enemies capability severity of response. Shortly after this first wave, a second wave of detonations might target densely populated areas and cities, as a means of reducing any rebooting capacity left over in the targeted nation.
In the aftermath of such a nuclear exchange, amid unaccountable death and infrastructure collapse all over the world the most immediate consequence would be the collapse of society. Without the people and infrastructure that keep society as we know it functioning, it would be highly likely that chaos would be the new normal, resulting in rampant violence and pillage of any resources still available. In an effort for survival, any all living people would likely be organized in what we might recognize as an armed gang or - at beast - an armed and closed community, which could live in a permanent state of conflict with each other as a way do procure whatever resources were available.
At the same time, radioactive fallout first and smoke and soot induced nuclear winter shortly after would settle on the land, compounding disease and famine to the remainder of humanity. Without the ability to produce food due to radiation and nuclear winter, the death toll in the first two years might be on par with other life extinction events eliminating the vast majority of terrestrial species and, perhaps, up to 95% of world population.
Depending on government preparedness and foresight, some pockets of civilization may have been gathered and protected in nuclear resistant bunkers, stocked with provisions to maintain these communities for a considerable amount of time in the hope that nuclear fallout would dissipate to manageable levels and nuclear winter would have cleared out.
In the long run, national borders will have been eliminated and the surviving humans will likely be organized in some sort of self sufficient city-sate. Global alliances will likely be irrelevant and substituted for local cooperation with near by city-states for which some form of trade is feasible.
As time moves forward, the better organized and resourceful cities are likely to aggregate near-by surrounding ones in what might be the beginnings of a civilization reboot, eventually resulting in the formation of new states. If this were to happen it could represent an opportunity for humanity to reorganize itself as one. However, given that isolation that city-states across the globe are likely to live in, I think the more likely outcome would to to foment local and tribal identity at first and nationalism afterwards.
In the end, if humanity survives such a devastating nuclear war, there is a high likelihood that in the long term, society would evolve much in the same as we know it.
Notwithstanding the above considerations, below are three predictions on Metaculus regarding questions that if they were to happen would likely push humanity closer to the doomsday scenario of total nuclear war.
The most likely scenarios for producing stratosphere reaching smoke but involve the detonation of one or more nuclear devices. This could happen as a result of a military or terrorist attack or as a result of a military test. I would think that a military attack is highly unlikely in the short term, mainly due to fear of nuclear retaliation, but would not totally dismiss the probability of a terrorist attack, if one or more terrorist groups could obtain a nuclear warhead.
In my opinion, a much likelier avenue for producing stratosphere reaching smoke would be as a result of nuclear testing, mainly by countries trying to assert their nuclear capabilities to the world.
All considered I would put the probability of at least one fire producing smoke plumes that reach into the stratosphere, by 2023, at between 10% and 20%.
In the event of an offensive nuclear detonation by 2024, which I feel is unlikely in the time frame of the question, I would imagine that the first targets would be military targets. However, once nuclear engagement is a fact, national capitals would be a likely clear target.
As such, I would put the probability of an offensive nuclear detonation on a nation's capital in the event of a nuclear confrontation at roughly one third.
In the event of an offensive nuclear detonation by 2024, which I feel is unlikely in the time frame of the question, I would imagine that the first targets would be military targets. However, once nuclear engagement is a fact, cities would be a likely clear target. However, given the destructive power and range of a nuclear detonation, repeatidly striking the same city seems redundant and perhaps an inefficient uses of the attackers arsenal.
As such, I would put the probability of an offensive nuclear detonation on a nation's capital in the event of a nuclear confrontation at roughly one tenth.