We have found AA flights direct from LAX to LHR, and BA, VA and AF (through Paris though) out of SFO. It's a long, long flight but if you can fly in biz I'd recommend it.
Best cards to get? We have used both URs and MRs for those flights (except the AA one) and I'd recommend going with them because they are much more flexible than choosing a particular airline. So, for international, cards like CSR or Amex Platinum if you want lounge access and Global Entry but don't mind the high AFs. You could apply for a CIP and get a good SUB, starting you on your way to collecting URs. Personally, I like to have a stash of both MRs and URs just in case one works out better than another.
As for the taxes, we have mainly chosen to fly AF because the taxes are cheaper, but then we weren't touring the UK. DD just flew VA through LHR last month on a last minute ticket to get her home and I think the taxes were less than BA but still not cheap (maybe $300). This was for a seat in "Upper Class" which was lie flat.
Hope this helps.