my uneducated take on what would happen post apoc
I'm guessing a majority of the deaths would be from the impending global famine. World population would probably take a fucking MASSIVE 'decline', in the billions of people. Like probably less than a billion people on earth in total, as diseases and other horrors run rampant after total societal collapse. I'm also that sure that a lot of governments would continue 'functioning' in some way. Probably governing in a decentralised fashion. Through radio and other ways of communication.
Also, how would we apply Marxism?
Honestly I think you'd have to go hunt down recent white papers on nuclear war theory. A lot of what people believe about nuclear war is based on vibes, cold war era tropes, popular media. The folks who actually drill down on fallout patterns, nuclear stockpiles, likely targets, effects on infrastructure and logistics, tend to look at it very differently.