Yes there are a few things BA crew are able to do.
One option is to sit with the passenger as if you are comforting them all though it is really just to put other passengers at ease. Not the task I would want to do.
The Singapore A340 has a chilled locker for use if a passenger passes away mid flight.
I am surprised the CAI flight diverted but I guess there was a hope they could save him but he must have passed away during the diversion?