I think it's inevitable that at some point in time something will break ECDSA.. I mean there's a pretty good incentive to find a solution to it so is a matter of when and not if.  QC or not.  The issue is how does cryptocurrency (and every other crypto related sector) survive.  It's pretty easy in my mind, it's no different than that bug where billions of bitcoin were printed once upon a time... it's a hard fork where new code fixes the issue.

So here's how I'd see it playing out.

Assumption 1.
Most likely it's the public keys in the chain that get compromised first as that's a lot easier than P2PKH style transactions where you need to derived a private key and script that matches the hash.

Assumption 2.
It takes a period of time to do P -> k solution (i.e. > 10mins).

With those assumptions we would likely get early indications of a compromise.  Early satoshi coins move, large accounts compromised to move the market, etc..
An update to the chain could be put out that offers a new signature system. 
If the attack is convincing that either a QC or fundamental crack has happened, a certain block number (even in the past) could be used as the reset point.

Everyone who wants to adopts the new hardfork.  Yes we have the BTC/BCH rubbish again but this is part of the bitcoin evolution process.  Strongest mining is 'the bitcoin' chain, so only through decentralised concensus is that reached.

Next anyone who wants to spend from an OLD tx has to also provide some PoW on their provided signature.  This means a simple GPU running for say a day does enough PoW to state they are the owner of the signature.  This would prevent even a compromise or QC attack from spending everyone's coins since they'd still need to do PoW for every tx they try to steal.

I'm sure there are even better ways to approach this but my approach is not to worry as we can always rollback and be inventive even on the worst possible attack imaginable.