You should have also mentioned whether you care about the seat much. Ie, if you're relatively short and any economy seat on an 11 hour flight is fine enough for you, then you've got a wider array of choices. If you're 6 ft tall like me and want a seat with more legroom on every flight, then you either need UA's Economy Plus (which I think you can buy instant access to, if you don't want to wait to qualify for UA elite), or you need elite access at any airline that lets you pick exit row seats with elite access, but to get to elite access quickly you either need a status match (which you can't get if you don't already have elite status with someone else) or at AA anyone can do a Challenge for either Gold (5K elite qualifying points on AA metal in 90 days) or Platinum (10k elite qualifying points on AA metal in 90 days).

Keep in mind that AFAIK only 4 airlines fly nonstop between LAX and London: AA, UA, BA, and Virgin. Virgin hardly partners with anyone, so that would probably not work for you. BA's pretty horrid in economy on those 747s IMHO, but they have a "premium economy" cabin (they call it World Traveler Plus) on this route which is about 7" more legroom (tho the same service as economy), and it's only in this "premium economy" class that you'll earn a worthwhile amount of miles (in discount economy, BA only gives 25% of actual miles!). UA as mentioned has Economy Plus if you're qualified to access it. AA has exit row seating (but only 4 exit row seats on the 777s they use on this route!) if you're elite (and as mentioned is the fastest to get to elite, on LAX-LHR you're definitely Gold in one round trip or Platinum in either one or two round trips depending on your fare bucket).

You probably want to fly nonstop eastbound as much as possible for maximum nonstop sleep time. However, on westbound you may want to sometimes include a stop if it gets you more miles , or if you're in a plain economy seat where it's nice to have a break midway. Also, keep in mind that for example at AA you'll be getting 4 500-mile upgrade "stickers" (they're actually electronic) for every 10k miles, whereas LAX-LHR is almost 11k roundtrip. But those 500-mile upgrades are only good on "domestic" AA flights (or legs of flights). Thus with an airline that gives you those kinds of domestic upgrades, it might be worth it sometimes to go westbound with a stop on the east coast so you can try for the upgrade transcon (or midcon from ORD at least).

One thing to be aware of BA vs AA: You cannot earn AA miles on BA flights US<->UK, nor can you earn BA miles on AA flights US<->UK. So you have to stick to one or other the other depending on which program you choose. However, you can earn or redeem AA miles on BA anywhere else (including most obviously London<->Sydney). And as soon as you get AA Plat with that Challenge, you'll have access to AA, BA, and Qantas lounges any day you're flying internationally.

In case you do want to upgrade to business sometimes (with miles), keep in mind that at UA you have to buy a more expensive fare to qualify for a mileage upgrade, while at AA you can upgrade from even deep discount fares (with very few exceptions) but you have to copay $250 each way. It turns out that adding the $250 copay works out to about the same as the higher fare on UA. But here's the thing: If your employer won't pay for the higher fare, but you're willing to pay the copay (if and when you use it) on your own dime, then since they're separate transactions that's something you can split between your employer and you, whereas at UA you can't.