Do the math.

Depending on your specifics, it may cost MORE $$$$ to use miles to upgrade to business to London on BA than on AA.

Here's why:

AA has only economy and business (and possibly first, depending on the plane).

BA has economy plus between economy and business.

To upgrade to business on AA, you buy economy and then add $250 each way, plus 25k miles each way.

To upgrade to business on BA, you have to buy economy plus, which is likely to cost AT LEAST $250 extra (over regular economy) each way, plus 25k miles round trip.

So you probably have to pay at least as much $$$$ to BA before you can upgrade to business as to AA. You just have the extra $250+ included in the ticket price (whether you upgrade to business or not) on BA -- and get exit-row style legroom in every row if you don't upgrade -- whereas it's a separate charge on AA now, only applied if you do upgrade.

Ok, well, you CAN upgrade from economy to economy plus with miles (and no extra charge) on BA, but all that gets you is more or less the equivalent of an exit row seat. If you don't qualify for an exit row seat (or frequenty travel with, say, kids who don't), or if you're having trouble reserving exit row seats on AA, then that may be worth it. But however better BA's business class may be than AA, I doubt many people would consider BA's economy plus to be better than AA's business class.

On the other hand: If you forget about $250 (because that's a wash), and focus on miles, it only costs half as many miles to upgrade round trip (from economy plus to business) on BA as (from economy to business) on AA.

And from California, you earn almost as many miles flying to LHR as it costs you to upgrade! So once you have 25kish BA miles in your account, you can recycle them for these economy plus to business upgrades forever, using Executive Club Dining or Diners Club transfers or whatever to occasionally "top off" your BA account (so you always have at least 25k miles there).