Let's pretend that a QC powerful enough now exists to break ECDSA within 10 mins.

Let's put on an "attacker hat" and see how it plays out...  

It's very unlikely an institution would do this due to it being illegal and the fine/lawsuits would ruin a company.  Therefore a single staff member is most likely to use a company's QC to attack the network... a bit like how people used to use company computers to mine with and later get caught.

The weaker, older "pay to public key" transactions would become the first victims.  Whilst this could include Satoshi's first minted coins, it's unlikely they would attack those early coins since any movement of long stored 50BTC tx's will always alert people.  Instead they'd focus on the most recent P2PK tx's and work backwards in time.

* Grab a P2PK tx and obtain the private key.
* Write down with pen and paper.
* Wipe or obscure the operation from QC history.

Manually construct the tx to another address, go to a coffee shop & VPN.. publish tx to a public home run node.
repeat, slowly and accelerate as getting more cocky that they got away with it.

So how is Bitcoin protected after such an attack?

This is the hardest part, we can't hard fork fix this since we wont know what is genuine and what is done by the QC.

Best fix I can think of is addressing the issues...
1. Add a quantum resistant signature system to Bitcoin
2. Require PoW for any transaction submitted to the node.

Ideally step 1 should be done now.   We don't need to use this new signature system but it should be ready to swap to.
Step 2... at a point when QC is believed to have compromised bitcoin, we require PoW of CPU power before accepting a transaction.  The transaction must pay to a QC resistant tx.
The PoW should be significant but not too much.  E.g. 1 hour's worth of PoW of the hash(nonce + signed TX).
The result of the PoW could be put in an OP_RETURN call or even a new OP code.

This way it becomes expensive for an attacker to steal too many coins and valid owners can use their laptops or mobile phones to issue transactions.

Hard to see how this would be rolled out in practice but P2WPKH happened so if there is a demand it could well be done.