It depends on how the the SHA-256 algorithm is broken. If somebody creates a quantum computer with 400 qubits and is able to create any arbitrary SHA-256 hash with any other prefix or expected "proof of work" with a Big O(1) level of difficulty, Bitcoins as a program would simply be screwed and potentially attacked as if somebody just brought on a million new machines into the network.
I did say "almost all cases". That would affect non-Bitcoin things, too. It'd be a case where everyone using SHA-256 is screwed.
In a worst-case situation, would the whole chain have to be restarted with a new genesis block, or do you think some algorithm from some arbitrary block number, say block 200000, would require the new hashing/cryptographically securing algorithm?
Apparently the NSA is trying to work on "stronger" hash algorithms and are willing to keep the effort to upgrade the algorithm as declassified material on the presumption that American civilians also need this kind of improvement to help protect network communications in general. Undoubtedly there will be stronger algorithms found in the future, and there may also be something which avoids the use of prime factorization and other approaches which are currently used for securing these hashes.
Yeah, it will be something that will impact far more than Bitcoins, that is for sure.