The Internet was designed with the idea of surviving a nuclear war, but bitcoin needs more than the Internet in order to survive so this is something difficult to know.

However a more important question is if human civilization could survive a nuclear war? Or even if humans as a species could survive a nuclear war? As not only people will have to survive the nuclear blasts but then they will have to deal with radiation, cold and lack of food, energy and medicines, which is an environment very few of us could endure.