Just for clarity, combining the information from all of the posts:

1. Irrespective of which combination of airlines you take -- whether AA/BA or BA/BA -- you will need to reclear security upon arrival at LHR.
2. If you take AA/BA, upon arrival at Terminal 3 on AA, you will follow the signs to Flight Connections to Terminal 5, which will lead you to a shuttle bus to Terminal 5. Upon arrival at Terminal 5 your board pass will be scanned and you will reclear security.
3. If you take BA/BA, upon arrival at Terminal 5 on BA, you will follow the signs to Flight Connections within Terminal 5, which will lead you to an area where your boarding pass will be scanned and you will reclear security.
4. Assuming you are traveling on a single reservation, in neither case will you need to claim and recheck your luggage at LHR, and in neither case will you need to clear immigration.
5. You will claim your bags and clear immigration upon arrival at GVA.