Transition for ECDSA is actually quite difficult. You face the conundrum of securing the older Bitcoin addresses, where they are vulnerable with P2PK or even those with the public key exposed. Quantum resistant algorithms are in the works now, but even if it broken the economics of it wouldn't make Bitcoin any more attractive than government secrets.
It was discussed in other topics, but in general, when it comes to ECDSA, you don't have that many options:
1. You can just introduce new address type, and do nothing with old addresses. This is what would probably happen, because it is also compatible with all other options.
2. You can burn old coins after some time. To make it compatible, you can use "do nothing" option, and make some huge pool for cracking and burning coins, that will be stronger than any single attacker.
3. You can redistribute old coins into miners. In case of any successful attack, miners will probably be powerful enough to take coins from any attackers, it is a similar case as with SHA-1 puzzle, where in practice only miners can safely claim those rewards.
Because it is hard to know in advance, what would happen, I guess people will keep status quo, as long as they can, so there will be option number one, and then if any solution will be needed, it will be just compatible with "do nothing" model.
Also note that speculating about the solutions now, can easily lead to totally wrong results. Would you expect things like "hardened SHA-1", before it was broken? Not really, people thought it will be replaced with a different hash function. But that was not the case, for example Git still uses SHA-1, instead of switching to SHA-2 or anything else. The world is simply unupgradeable, and relies on soft-forks everywhere, if something is "legacy". So, in general, I think no matter what will be broken, the actual "fix" will handle only this particular attack, nothing more, nothing less.
If SHA-1 was turned into "hardened SHA-1", then I expect secp256k1 could be also replaced with "hardened secp256k1". Even if private keys will be reached by the attackers, there are still many options, like "a proof that some key is a part of some HD wallet". The same with signatures: if they will be broken, then you can force using deterministic ones. The fix will be highly dependent on the attack, for that reason we don't know right now, how exactly it will be fixed (because today we don't know how the attack would look like).