If SHA-256 became completely broken, I think we could come to some agreement about what the honest block chain was before the trouble started, lock that in and continue from there with a new hash function.
So would the world stop all transactions while a patch is developed and put into place, then once that's done, ask everyone to recreate their transactions from the point of discovery onwards? Let's say it's a large organisation and they put through 10,000 transactions an hour - that could be a lot of work to redo.
If the hash breakdown came gradually, we could transition to a new hash in an orderly way. The software would be programmed to start using a new hash after a certain block number. Everyone would have to upgrade by that time. The software could save the new hash of all the old blocks to make sure a different block with the same old hash can't be used.
Let's hope quantum computing theory doesn't become reality then.