Quite a lot of the USSR should have quite the bone to pick with Moscow after Politburo started WWIII. Even moving army units to quell dissent is going to be a problem with the logistic and basing system literally blown to pieces.
I would not be surprised if as many of the people that can afford to move elsewhere just leave. Not much point in trying to rebuild a ruined country when round two between nuclear powers could kick off again and your stuck in the middle. The border should be leaking refugees like a busted dam.