An addendum to my previous post: I checked the British Airways website to get a more complete picture of the travel situation. It was there I found the attestation form you'd be required to fill out:

https://www.cdc.gov/quarantine/pdf/Proof-of-COVID-19-Vaccination-For-Noncitizen-Nonimmigrants-Passenger-Disclosure-and-Attestation.pdf

One of exceptions listed is humanitarian or emergency travel. You can verify with the airline, but I'm nearly certain a death in the family constitutes as emergency travel.

Now here's the thing: On the attestation form you have to pinky swear you're going to self-quarantine for 5 days and get tested, and if you are staying more than 60 days you're going to get vaccinated. However, to return to London you do not have to provide proof of being vaccinated, and I don't see anywhere that you have to provide proof of the stuff you agreed to on the attestation form. You should ask your airline or your passport agency if proof that you did the stuff you agreed to on the attestation form has to be provided to anyone. If yes, then I would recommend against traveling. If no, then it is up to you. The airlines and passport agencies may have a better idea of how it all works, but you do want to travel for emergency reasons, and fielding questions about these types of scenarios are common for these folks to deal with.