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.