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If your primary mileage goal is to redeem for TATL award travel in Coach during times of the year when you can actually find off-peak SAAver award availability on AA/US metal, using BA Avios would yield relatively poor value because of the hefty surcharges BAEC imposes on such redemptions. If you redeem AA miles instead, you won't pay those surcharges if you fly on AA/US metal.

If you will have to buy at least one TATL ticket for cash, make it the one for yourself, and fly on AA/US. You don't say what your U.S. destination will be, but even if it's a place with nonstop service from LHR on AA, you don't necessarily have to book the nonstops. If you book a connection in each direction -- say LHR-RDU-LGA/JFK-RDU-LHR, instead of LHR-JFK-LHR, you'll get your four AA segments, and might -- or might not -- pay less than you would for the LHR-JFK-LHR nonstops.

If, one the other hand, you were looking to use miles for short- or medium-haul travel on BA from LON to the rest of Europe, BA Avios would be far more valuable than AA miles, because of BAEC's distance-based award structure, and the availability of Reward Flight Savers.