Well technically it could work. However, take a look at:
One round of the algorithm takes 16 minutes, 45 seconds which works out to a hash rate of 0.67 hashes per day.
Unless there we a emergency fix that would lower the difficulty, this would be pointless. Additionally, do you really think that someone would worry about Bitcoin mining in a 'massive nuclear war'?
What about nodes? how do you check funds and validate transactions?
If there were nodes with connections, they would still work. Technically you don't need mining for nodes to operate (albeit you won't have confirmations).