As a BA 747 captain, perhaps I could add my tuppenceworth. I am not a manager and have no other knowledge of this incident than any other member of the public.
1) Whatever else I may think about British Airways, and I am happy to take it on the chin wrt service, punctuality etc etc, we do NOT compromise safety. When the decision was taken to continue to London there will have been NO question about the safety of so doing.
2) A 747-400 on 3 engines, at the sort of weight it would have been out of LAX, is just as safe as any twin-engined aircraft. I.e. it has FULL redundancy of ALL systems. As with any twin-engined aircraft it can cope easily with any engine failure, including the one on the same side. ALL hydraulic systems (and therefore controls, flaps, gear etc) will continue to operate normally.
3) A 3-engined 747-400 will cruise at a lower altitude and will therefore burn more fuel. The Flight Management System automatically calculates the new fuel burn on 3 engines at the lower altitude and presents the pilots with an accurate estimate of fuel on landing, whether at the intended destination or an alternate.
4) I have no idea why, towards the end of the flight (as I understand it), a 'Mayday' was transmitted. What I can say is that the transmittal of a 'Mayday' will guarantee that you will get a priority approach, so you have no concerns about whether the aircraft ahead will burst a tyre on the runway, be slow to clear the runway, or whatever......because the runway will be maintained clear for you to land on.
In short, the continuation of the flight is a non-event and the declaration of 'Mayday' ensured the continued safety of all on board. It was not an indication of imminent disaster.
It would be a real shame if all the publicity given to this event were to pressurize me or my colleagues into not transmitting a Mayday if necessary, because of all the publicity it might generate (17 pages so far on this site, a full page in the Times and God knows how much elsewhere).
Rossma - I deeply feel for you and your family. As I say, I have no more knowledge of what happened than most other people, and certainly far less than you have. I can say though that the guys in the flight deck will have been working flat out to sort the problem. It is always a balancing act between solving the immediate problem and keeping the Cabin Crew and passengers 'in the loop'. If there's a conflict, flying the aircraft will - and must - come first.