When The Current Ex-Mrs Spike was due to start chemo around the time we were supposed to be taking a BA holiday we were at first offered the chance to transfer the amount paid to a future booking. I know a pregnancy and chemotherapy are not the same things but a bit of pleading your case may help.
After the initial offer to carry the deposit on to another booking I reached out to BA via Twatter and ended up getting a full refund.

Spike