I think the community will replace it with a better algorithm and existing coins will be swapped for the new coin. Transactions might be required to be frozen for a while though..
How do you make sure that only the original owner of a P2PK can swap for the new coin, and not one who also found the private key? Isn't that private key the only thing that can be required for authorizing the swap?
You clearly don't understand. Once the BTC cryptography is compromised, there is now way to prove chain of custody, so there is no way to prove who has how much bitcoin and assign it to a new coin. Literally, in an instant, the price of BTC goes to zero.