Sorry to hear and wishing you & your family long life.

BAH should use the same policy on Compassionate issues, meaning a full refund (they may ask for proof but normally they dont).

If there is pushback, one thing to note is the rules are different Pre-Departure opposed to once you have commenced your journey - so absolute worst case you might be able to use this and imply that you'll fly INV-LHR but then refund. This would be a colossal waste of time and effort for all involved, let alone the emotional factors involved during this challenging period.

BA tend to be very helpful in these sorts of situations.

M