Even if SHA-256 is partially broken (say easier to find hashes, but not fully broken), you-d see major resistance from ASIC miners. You might even get a split network.
SHA-256 wouldn't really need to be swapped out unless it was fully broken. The most significant risk of a partial break would be ECDSA with the secp256k1 curve. The miners really shouldn't care much if this algorithm were swapped out, and all non-mining bitcoins users would have a financial interest in accepting the swap. If a significant number of users (10%?) started using a client that used something other than ECDSA with the secp256k1 curve for signing transactions, the miners who wanted to collect the fees from those transactions would quickly accommodate the inclusion of the transactions in blocks.
I don't believe this change would even require a fork, but if it did I don't think that would be a significant hurdle after a partial break of ECDSA with the secp256k1 curve is discovered.