This is BA, a commercial private airline, not the NHS; what right do you have to expect compassion. If you had enough foresight to know you might have gotten sick, you'd have bought a fully flexible ticket for 4x the price! Got it!?
Seriously though, this is an area where BA (and I would add some other European carriers) do fall behind on; here in the States and even in Asia for that matter, getting a same-day change is rarely an issue, especially with a real reason. Even without you, you can typically pay something nominal for a same-day change if you want to go standby (and usually free, with status). I understand that BA would be worried about losing revenue from business folk who arrive early and want to catch the earlier service but customer service isn't a one size fits all ordeal, compassion and understanding is what gets people back to your service.