The claim that quantum computers will break Bitcoin appears regularly and is usually stated without specifying what breaks. Two different mechanisms are involved and they are affected differently.
The signatures are the vulnerable part
Ownership is proved with elliptic curve signatures, and that is the component a sufficiently large quantum computer would break. Given a public key, it could derive the private key.
The important detail is that the public key is only exposed when coins are spent. Addresses that have never been spent from do not reveal it, which means a substantial portion of holdings are not immediately exposed.
Coins in addresses that have been spent from before, including very old holdings, are exposed.
Mining is affected much less
The hashing that secures blocks is a different mathematical problem. Quantum algorithms provide a square-root speedup rather than an exponential one, which is a meaningful advantage and not a break.
The timeline
No machine close to the required scale exists. The requirement is millions of physical qubits with error correction; current devices have thousands without.
Estimates for when this becomes practical range across decades and are, honestly, guesses.
What can be done
Signature schemes resistant to quantum attack exist and are standardised. Migrating a blockchain to one requires network-wide agreement, which is politically harder than technically.
Coins in lost wallets cannot be migrated by anyone, which creates an awkward situation nobody has a good answer for.
The reasonable position
A genuine long-term risk being addressed slowly, not an imminent threat. Anyone using it to argue for urgent action is usually selling something.