If you are flying Business Class (J) long haul, I believe you can get access to the BA Galleries lounges in LHR. I am not an expert on the BA lounge access rules, so I certainly invite correction from those who are. I believe, also, you can get access to the lounge in TBIT in LAX. Since I always fly AA out of T4 in LAX, I am not familiar with the TBIT lounges.

There are two BA T5A lounges, plus one in T5B. One at the north end of the terminal, one at the south end. When I flew J into LHR from LAX, I simply wanted to know which lounge would be closest to my departure gate. That's why I tried to figure out, in advance, the probable gate. As I mentioned before, that plan was unsuccessful.

Last time I just asked the guy who checked my BP before the escalator up to Security and he indicated that a A22 was "penciled in". Thus, I went to the South lounge and, luckily for me, that "penciled in" gate was in fact the departure gate. I had a relatively short walk to the gate.

With AA J tickets LHR-LAX, you can certainly get access to the Admirals Club in LHR T3 on your way home. I believe you can access other OW lounges, as well, but I personally have never done that. The AC is fine for me. Some say other OW lounges are better, for what ever reasons. I'm not all that fussy.