I connect regularly at LHR and have tried them all. I would rank them BA > AA, with advice to forget CX. As others said, BA has the edge in food and drinks, AA in showers. You'll have to keep track of time yourself; don't count on announcements. I use the alarm function of my wristwatch or mobile phone (the alarm works even if SIM card doesn't).

If you were alone with enough time, I would have advised you to lounge-hop and post your own comparison, but I understand that this may not be practical with family. (BTW, to avoid any inadvertent misunderstanding, the advice to forget CX only applies to LHR. Their lounges in HKG are fantastic.)