One final question: how badly was the atmosphere messed up in terms of affecting the sunlight? Was it, like, full nuclear winter for a while, or not so intense?
Good question! TLDR: Climate was falling apart beforehand, then there was nuclear winter and storms for like 2-3 years, and the climate is still super unstable but is just now starting to find an equilibrium.
Long version:
The climate was already jacked up prior to the war: the Gulf Stream current had totally collapsed, plunging the northeastern and Atlantic U.S. (as well as Europe) into a minor ice age. This actually became a point of contention between the United States & Europe and the rest of the world: as the rest of the planet suffered from deadly temperature rises, the western countries increased their reliance on and consumption of fossil fuels.
In the Gulf of Mexico, constant hurricanes and a dramatic rise in temperatures thanks to a constricted Gulf Stream turned the region into a steaming, flooded, stormy swamp hostile to human life, but with plenty of tropical creatures and mutants making the place their home—especially once the humans left.
Out west, everything began drying up thanks to a climate-change fueled El Niño—forest fires and dust storms ruled the day, punctuated by the occasional superstorm from fluctuations out east or from the Pacific. Even Seattle now has more clear days than rainy ones.
Between these three regions, starting in the Blue Ridge Mountains in the southeast, stopping at the Rockies in the west, and stretching all the way up to the Arctic, became an unpredictable nightmare of floods and droughts and storms of all kinds.
And the kicker? All of that was happening
before the nukes fell. Needless to say, the large-scale disasters and shifts accelerated the societal collapse. Tensions over food supply, fresh water, energy, climate policy, critical resources, and the shifting resource landscape meant that nation states found themselves at each others throats much more than they had in the past.
When WW3 did break out, there was a full-on nuclear winter for about two or three years. This was characterized by a dimmed sun and loads of precipitation year-round, fueled by the ash and dust shot into the sky by the war: cold rains in the summer, blizzards in the winter, thunderstorms and dust bowls whenever. The global temperature dropped by quite a bit and remains depressed, although there are indications that a rapid rise is in the way, possibly even outstripping the previous rate of global warming…
Currently, things in the US are as described above, just much colder and with significant variability from time to time: the west is dusty and dry, the center is chaotic, the south is hot and wet, and the northeast is frozen. In Europe, things are also frozen, and in the rest of the world, things are hot and swing back and forth between flooding and drought.