I haven't seriously looked at Bitcoin's PQ plan for a couple of years, so I might be (I am almost certainly) out of date, but my recollection is that there's a "pre working attack" phase required, in which everyone basically signs a new PQ secure address, and a cutoff date.This would leave holders who did not sign in two categories:
1) If you never sent a tx with an address, then you did not reveal your public key, and have some safety, e.g. you could do the PQ signature, wait, and be fine.
2) If you did, then you revealed your public key, and didn't bother to make the cutoff, and well, too bad.
There was a bunch of frankly dumb analysis about how long this would take the chain to process and how expensive it would be assuming that miners would all continue to enforce 10 minute blocks and transaction fees for these signature txs. I would be very surprised if the mining industry shot itself in the foot like that. The actual time to process 200mm or so new signatures just isn't that long. Hey we could do it on Solana if we needed to. That said, I imagine the papers this week plus Google moving up its timeline mean that there will be a concerted effort in Bitcoin land to get a real process down and tested in the next couple of years. Pretty cool.
Finally, I've read very little analysis about whether or not miners would choose to continue the energy dependent nature of mining, or try and move on. I think this is a pretty interesting economic question; I'm looking forward to finding out the answer. I expect mining will have a longer lead time than the signature problem - we're a long way from having Grover implementing SHA-256 as far as I know. And even then you still have 128 bits to deal with ONCE you get an equivalent amount of Grover-capable quantum compute out to the current ASIC ecosystem.