we are flying British out of Paris and American out of London.
BA and AA are code share partners. If you haven't bought your BA tickets yet between Paris and London Heathrow (LHR), once you do, contact AA and request your trip be put onto 1 itinerary. If you get an agent who says no, call back and speak with another agent. I know Delta will accommodate this request; I would think AA would too. One itinerary would give you some protection in case your BA flight into LHR is late.
With that protection and if your AA flight departs FROM THE SAME TERMINAL AS THE INBOUND BA, if you don't want to spend the night at Heathrow, you could fly into LHR 4 hours in advance to allow for passport control and security. In the same terminal, you shouldn't have to go through immigration. I speak from experience in that I flew from the JFK to GIB via LHR on two different tickets and didn't go through immigration. I didn't have to leave Terminal 3 and I gave myself a 4 hour transfer time cushion.
So just an idea. That 2pm flight gives you flexibility unlike an early morning transatlantic flight which is nice.