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.)
I’m not saying it will happen but it is plausible. The question is what is the effect of that happening to bitcoin.
If someone broke SHA-256, they could generate blocks directing all coins to their own address and thus destroy the entire system. However, in that scenario it is likely that everybody else would roll the chain back to before the break and restart it with a different algorithm (likely SHA-3).
In summary, an obvious compromise would get caught and neutralized.
So a wily possessor of an SHA-256 break would use it subtly. Most likely they would target “lost” coins that haven’t moved in decades that nobody will notice are missing. Not Satoshi’s though. Too much heat.