The rules on access to the the BA lounges are shown here:

https://www.britishairways.com/trave...club/_gf/en_gb


Note that access to the Arrivals Lounges is determined by the type of ticket you have arrived on, not are planning to depart on. So you need to be arriving on BA long-haul in F or J to get access, or in any class if you are BA Gold.

Connecting AA-BA, you can either head straight to the BA departure lounges (T-3 to T-4 is by bus airside, T-3 to T-1 is also by bus airside, unless you want to go landside and then it's a 10 minute walk), or you can go to the AA Arrivals Lounge and transfer to T-1/T-4 landside.