Basically, if there's a war, most people will die, do you think they're going to end up thinking about their cryptocurrency investments? No!
The importance of cryptocurrency at that time will actually fade and the market will fall. All of the people will just worry about the biggest problem which is the war.
Okay, all this was my view only when the US and Iran have a massive war so what's yours?
During World War 2, gold was in huge demand as it was portable money that was accepted everywhere (whereas fiat from one country might not be accepted in another).
Many jewish people got out of Germany carrying some gold on their person - they were forced to sell their property for a fraction of it's worth, so basically in a crisis like that, gold was the only asset worth having because you could hide it and carry it with you.
I'd say bitcoin functions in the same way (in that you can exchange it anywhere in the world).
Regarding "the internet going down" - it was designed to work even if some cables get cut off and even if electricity is down in some areas. You would need to take down electricity all over the world and cut the cables all over the world, for the internet to go down. Which is extremely unlikely.