Unfortunately I went through something similar a decade or so back. Slightly different as the onward flight was to NCL and we only got the news while in transit, but I simply found someone from BA, explained and they offloaded us.

We made straight for Hertz, picked up a car and headed to hospital. A few days later I drove back to Heathrow and collected the bag from the designated location.

Do not wait for your bag at Heathrow as it may take them some time to pull it from the system.