My condolences to the OP.
When my Dad was suffering from cancer, I was flying weekly from the DC area to LHR and once BA found out (from a CSD I had a conversation with), I had special services look after me each time I was at LHR along with staff greeting me at either IAD or BWI. Totally over the top but really appreciated.
Had two tickets booked for my kids to come see him, but his health had deteriated to the point where it wouldn't have been a good idea. I sent a note from his doctor explaining the situation along with this death certificate and BA refunded their entire non-refundable flight with no questions asked.
IMHO, BA does a fantastic job when their customers are going through hard family times. Whilst having a fag outside T5 once during this time I saw WW waiting for a car and went to thank him - his response was simple - we look after our customers during hard times. Bravo BA.