If a nuclear war really comes, what we should consider is whether we can still live? Will there still be a country at that time? Whether there will be human beings is a question, but I think at least there will be no nuclear war in the few decades we are alive Because although there is a war at present, it has not reached a very large-scale war, and human beings have not formally contacted alien civilizations, and cannot immigrate to other planets, so what is the benefit of launching a nuclear war now?
You may be will surprise but no wars are beneficial for humankind so there are some other reasons for them always, most are not really wise. If we talk about nuclear war it is totally mad, but we know there are mad people in society. So some insane things do unfortunately happen.
Bitcoin itself will survive as while at least one copy of its ledger exists Bitcoin exists. So main question is if a society will be advanced enough to have need and opportunity to use it and there might be many variations.