Actually once you soft fork to add the quantum resistant code, you might also add something to keep those coins locked forever. Something along the lines of: By the date the new code activates, coins not moved to a new wallet that supports it become locked. You can wait a year or four (next halving), etc. to give people plenty of time... Well we will see what core brings regarding this matter.
If you keep those coins locked or something else what changes the original supply, you will create another sh**coin fork.
The original chain will always remain the main chain with all the mined coins. Bitcoin is a challenge network. There will be a quantum update. But whoever moves the coins with no owner from the old to the new addresses will be the new owners. That is what original Bitcoin is. That makes the community improve the network.
SHA-256 is not a quantum issue but to have an example:
SHA-256 is very strong. It's not like the incremental step from MD5 to SHA1. It can last several decades unless there's some massive breakthrough attack.
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.
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.
"Keep using the longest chain" or maybe "use this chain instead from now on"
I did a poll earlier a few months ago I think 19 out of 20 people said they would use whatever satoshi said.
Thereafter we would have two chains. The original chain and a forked one. To avoid such situations the Bitcoin community has to have an updated network all the time. That is the challenge.
The quantum update will mark the new beginning of Bitcoin as the lost coins will be active again.