On the mining side, miners would pick a new algorithm, vote, and then cut over when approved. Although, if a post-quantum technique allowed faster mining, a change to the mining algorithm may not be needed. Bitcoin would notice the faster hashrate and increase the difficulty, miners would rush to buy quantum hardware. As long as quantum hardware is generally available for purchase, we'd still have decentralized mining.
On the wallet side, we'd need miners to vote for and approve quantum safe wallet operations. Users would need to update their software and their wallets. I'd expect some stuck wallets (like the initial blocks, lost keys) could get spent. This is sort of like someone recovering gold from a sunken ship.
I agree this seems like an existential threat, but there is an update path. Unless there's a sudden onset where no-one has time to react, I suspect the miners would reach consensus. Some users may be slow to update their wallets, and may suffer losses.
I think we'd eventually see recovery of the coins from the mining of the initial blocks, which are generally believed to lost/stuck. This would sort of be like recovery of gold from a sunken ship. I've heard that recovery of those coins would destabilize Bitcoin value, due to the large value. However, unless they are all unexpectedly recovered at once, I'd expect the market to "price that in" gradually as recovery begins.
I'd expect many cryptocurrencies not to survive the transition, although I'm a crypto skeptic and I don't expect hype/scam coins to survive anyway.
[1] https://blog.coinbase.com/bitcoin-s-elegant-upgrade-mechanis...
One thing seems certain, waiting for NIST is folly.
https://stribika.github.io/2015/01/04/secure-secure-shell.ht...
https://csrc.nist.gov/Projects/post-quantum-cryptography/rou...
"Formal complaint regarding 8 June 2021 incident - 2021.06.15, Daniel J. Bernstein
"Executive summary. A week ago Dr. Daniel Apon from NIST publicly accused me of professional misconduct. Specifically, he accused me of initiating private contact with NIST so as to provide false information to NIST regarding the timing of an upcoming announcement relevant to NIST’s ongoing decisions..."
https://www.openssh.com/releasenotes.html
There is a non-zero probability that a quantum computer can break these key exchanges.This might not require a full application of Shor's algorithm, as studies continue with statistics from quantum annealers.
https://www.forbes.com/sites/arthurherman/2021/06/07/q-day-i...
If this happens, cryptocurrency as currently designed collapses.