I couldn't care less about Bitcoin's use of SHA-256, but based on experience:

First of all, it's quite unlikely that this is going to happen at all. There are no signs of significant weaknesses in SHA-256.

But if it's going to happen, in all likelyhood, you'll get preliminary results long before any actual breakage. (For both MD5 and SHA1, you had around a decade of warnings between "this looks insecure / broken in theory" and "we now have an actual collission".) Anyone paying attention and moving away from algorithms already known to be broken was unaffected by the MD5/SHA1 weaknesses.

Of course, there will be those who will wait until an actual breakage, and a few years more, before they act. (As we've seen particularly with MD5.)