colinAMS , since you will be starting from EDI, GLA, or MAN, you'll probably best off going for status with oneworld (AA, BA, or IB, perhaps).

For choosing flights, EI might have been a good option, either via DUB which has US pre-clearance and TATL nonstops on both AA and EI, or on EI's MAN-JFK nonstop. But EI does not offer Premium Economy. (AA does.)

Status with AA would possibly get you upgraded on AA domestic flights, but not lounge access on a wholly domestic itinerary. Status on BA/IB would get you lounge access when flying AA on a wholly domestic itinerary, but usually no upgrades when flying AA domestically. Status on AA, BA, or IB would get you free Main Cabin Extra (extra-leg-room) seats and free checked baggage when flying in Economy on AA -- even on AA's cheapest "Basic Economy" fares.

AA miles are more valuable than Avios for many types of flight redemptions, but it sounds like you would not be using your miles for flights.

Which U.S. cities do you expect to be your most common TATL destinations? How many of them are served by BA from LHR, or by AA from DUB/LHR? (I note that AA also offers summer seasonal service on the EDI-PHL route.)