It may depend on how many nukes fell and where. Riker's statement in First Contact that most major cities were destroyed and there were few governments left could still be very much valid if civilization basically collapsed during the Post-Atomic Horror. It only takes one or two good-sized nukes to completely devastate an entire country--and not everything has to be burned to the ground either. Historic landmarks like the Golden Gate Bridge, the Eifel Tower, and that gigantic ketchup bottle in Collinsville, Illinois could still be left standing if society and towns fell into ruin around them, IMO. It could be that many cities were abandoned and left to fall to seed as people fled to various sanctuaries and makeshift settlements for survival.
Given time, the aforementioned landmarks would probably decay and fall as well due to a lack of maintenance, but if some reconstruction began only a few years after Cochrane's flight, they could have been saved.
The destruction of major cities would be a blessing in disguise, honestly, since it means a lot more farmland and so would be more unaffected, allowing for farming and small towns and bunkers to emerge and rebuild a base asap. I think the deepest of the British documents that Thatcher buried basically wanted the urban population to stock up and stay put. If they die, less people to worry about, if they survive, they'll have enough to huff it until forces arrive and lock them down for a harvest or two hopefully (and keep dying off in the cities, and leave the farmland alone).
But a big thing is most people, especially in the 80s, gave every which what prediction for nuclear war, majoring into the fear mongering. They also tend to underestimate or outright forget the global south, which yes, while tied to the north for food and finished goods, isn't wholly incompetent, especially in agriculture and each nation having some corps of technical personnel...
Sagan and Nuclear Winter seems to be a wrong conclusion, as Sagan admitted after the ecocide in Kuwait in miniature. This does not mean there wouldn't be cooling, but just that human technological civilization, and most importantly, our agriculture, is more likely than not going to survive an exchange.
We also don't know how many nukes went off or their yield, just enough to kill 2 billion, which if you incinerate the major cities, wouldn't be amiss but still leave 6 billion around. At worst, half of those might starve before the carrying capacity and agriculture recoups, less if nations were actively working on stockpiling fertilizers and farming equipment et al. Or, even, in trek the 2 bil is both those killed in the exchange plus the rough number that died due to the collapse afterwards, in which case woo! What a light nuclear war. Especially if stockpiles were light in number, even if their individual yield kills a city, that's still better than the utter saturation the 80s stockpiles threatened. Add: it also seems that Trek implies it's Econ, not the West/NUN/Nato, that takes the hit more...if India and China are taking most of that 2 billion, while still, of course, a tragedy, it sort of regionalizes the damage. The US and Europe seem to get on their feet fast enough and be big players post-war, after all....