The thing that supposedly sets Bitcoin apart from other cryptocurrencies is that it's deflationary and 'immutable', in that Satoshi is gone forever and any deviation of Bitcoin from his golden idea will result in undermining its essence. If Bitcoin can get quantum-attacked then, from a technical point of view, nothing will be lost. The Bitcoin core devs can issue a word-of-god statement stating that they'll roll back…
Thus, upgrading the protocol/code to change the encryption to something quantum-resistant should be no more controversial a change than segwit. The community has already answered the "is it still Bitcoin". Yes it is, protocol and code is free to change given longest-chain consensus.
The problem will be what to do with legacy addresses. Never before have issued coins been forcibly deleted by a BIP. It could turn out that legacy addresses (including Satoshi's) that fail to have their coins moved after a deadline must be considered compromised and burned/destroyed. That has no precedent with bitcoin, although it does with ETH.
Anyone know if there's a way out that doesn't require this? Obviously there's no way to ensure all legacy address coins are moved by the deadline.