You will need to clear immigration to use the BA Arrivals Lounge, but you will not need to pick up your luggage. I recommend that you use the arrivals lounge, as it is superior to the AA lounge. When you get there, immediately see if you can get on the schedule for a chair massage. Then, you can have breakfast, use the internet, and relax.

As long as BA has given you your boarding card for the AA flight, you only need to head to Terminal 3 about an hour before your flight.