On the assumption that you're going economy, I would definitely pick BA or VS over anything hailing from the other side of the Atlantic.
As between BA and VS, it's mostly a matter of taste. BA (in my view) is consistently good overall, but it's more restrained/refined/sophisticated (take your pick) than VS and some people find it a bit cold and stand-offish. BA a/c have virtually all now got personal seatback TV screens, thus nullifying one of VS' main selling points for most pax. (I hardly ever watch anything on an a/c so this has never been an issue for me.)
VS also sounds like it's been suffering a bit from cuts, so some of the goodies have gone. My main problem personally with the VS product itself is that the entire cabin service is geared to the inclusive tour market, and it can be intrusive and annoying if all you really want is to be left alone to your own devices unless and until you want some service. There are also some reports that the cabin service on return flights to the UK is less good than on the outbound flights because the crew is, er, less fresh and rested. Nevertheless, I have friends who fly both BA and VS who would pick VS every time because they have more fun.
To LAX, you would also have the choice of flying on Air New Zealand. It used to be really good, but what I have heard lately is that service levels there have suffered a bit too. Must still be better than the AA/UA bus services that masquerade as airlines.