My experience with BA and BMI when my grandfather died was fantastic.

My parents were visiting my aunt in Zurich. BA changed their flight at no cost at all, didn't ask for any evidence, and charged very reasonable walk-up fares for my aunt, uncle and cousins.

BMI sold me a Prem Econ ticket from TLV at four hours notice with a flexible return date, for about £400 IIRC (this was 2009), again just on my word that it was a bereavement, and gave me a courtesy upgrade to Biz on check-in, same on the way back (may have been a coincidence and I was BMI Gold). They had clearly also made a note for crew, as the CSD came up to make sure I was okay.

The whole family was back in London by that evening at minimal cost and hassle. Very impressive on all counts.