Everything comes to a stand still when there's a war so that wouldn't be different if a Third World War was to broke out. Bitcoin will suffer in such a circumstance just like any other thing or systems in place.
of course bitcoin would suffer, but it would become an even better alternative for the cash or fiat that we are having right now, because bitcoin needs only internet and nothing else.
The war back then would be different from the war that's going to happen in the future. Back then, we have tanks, guns, and all those stuffs that you typically see in a movie. But today, we have nuclear bombs, and other deadly/lethal weapons that can destroy a whole nation. Probably most of the ISP towers would be already destroyed after the next war. Since bitcoins need the Internet, it would take a couple of months before all those towers will be repaired.
Another thing, people would be waiting for the electricity lines to be fixed. Since you need electricity to charge your phone/laptop, you wouldn't be able to use bitcoins at that time. Probably the same fiat will be used by most of us right after the world war. After a couple of months, there might be a problem with a nation's fiat so they'd choose to use bitcoins instead.