The only idea I had was to try get on to a morning flight out of JFK, but they are both full and you'd need to make the journey very very early for the BA anyway

You don't say what class you are in but the scheduled BA66 is also very full - just 1 x J and 1 x F seat available. So I think realistically you'll just be waiting this one out - hope you get away OK!

Should be a simple EU261 claim - check out the threads on here