No evidence that this will happen if you're talking about the hypothetical civilization-ending nuclear winter (in fact very unlikely), unless you mean that both Russia and America will being targeting random cities all around the world just for funsies? Utterly moronic, why would you waste nukes on random innocent people and innocent countries instead of your adversary who is trying to make you extinct? Silly. Not to mention they simply don't have enough warheads to cause the end of the world through targeting all cities.
Go check up on the yields of the vast majority of bombs then actually plot them:
So the only way for a nuclear exchange to cause the end of the world is through the hyped up nuclear winter theory which as many smart non-political people have shown is very improbable. You can red some snippets on the controversy of the hyping up of the supposed end of the world through global thermonuclear war (perhaps its good liberal academics are lying this time) here:
