Let us not to talk about war so slightly, it is a serious manner and we all are supposed to avoid it...
OP should understand that if worldwar3 happens that a currency that'll replace US dollar, will be the least of his problems, because there'll be so much destructions that we might be thrown back to medieval or stone age. As you said, the level of weapons of mass destruction that's available now is capable of mass destructions that the few people that'll survive Will probably be living on gas masks. I doubt that there'll be any need for financial institution for a couple of decades, because as you said again, there'll probably be no stones over stones. Where will you buy and sale to warrant financial transactions, I believe that money will become worthless, because people will only be interested to hurdle food items to survive. So let's be careful what we wish for, because the aftermath might be horrible.
I acknowledge OP and nobody in this thread is actually wishing for a war to happen, I believe what they say is mostly attributed to morbid curiosity, still curiosity is not enough for me to dive myself in an exercise of imagination which will likely only end up with me suffering from fear of what the future awaits for all of us in this era when all the major power worldwide have access to weapons of mass destruction. Not only talking about atomic weaponery, which is certainly the newest, but also chemical and biological agents which will likely not leave any human being alive in entire countries.
It is rather scary and silly to even believe there would be a chance for the currency financial institutions to survive such scenario. The stock market would cease to exist, private companies would lose all their assets and governments would go so desperate that they would likely seize the wealth of corporations in order to survive as long as possible.
So,instead talking about the outcome of a war which would leave no winners, let us talk about the things which are supposed to be done, so those horrible things won't ever happen in the future.