From which city would you be starting your journey?
I fly United a lot (about half as much as Geo), and imo they're absolutely fine and improving; their flight attendant service improved dramatically about 18 months ago and is maintained.
Seat wise, regular economy is 31" pitch which is the same as BA and LH (Swiss has 32"). Apart from the 747 (which you'd only get if flying to San Francisco) all their international fleet have in seat video. Of those, all** except the 3-class 767 have AVOD. Food is economy class food, sometimes good sometimes bland, so take some snacks. United does charge for alcohol on transatlantics, and you pay by card; use a credit card for this as some UK debit cards charge a transaction fee. You can also select your seats before check in without having to pay (BA charges £25 per person per flight).
If you wanted to, you could also upgrade to Economy Plus, which gives you a much better seat pitch (35", that's 4" extra legroom). Those charges are around £70-80 per flight (plus extra for the domestic sectors), which would wipe out your cost savings but get you a much better seat than the others.
The advantage of flying BA (assuming you'd be flying into Vegas) is that you'd do US immigration and customs at Vegas, and not at an intermediate airport en route. If you fly United, Lufty or Swiss, allow plenty of time to do that at your connection, I'd say 3 hours.
(** - yes there are still six 3-class 777's without AVOD, but they're flying the second tier Asian routes)
Edited: 6:12 pm, March 02, 2013