I would have no confidence in a layover of only an hour at Heathrow. To begin with, you're almost certainly going to need to change terminals. Long-haul AA flights arrive in Terminal 3 (https://www.heathrow-airport-guide.co.uk). All BA flights operate out of Terminal 5. To get from Terminal 3 to 5, you need to take the Heathrow Express train service. This service runs every 15 minutes, and estimated travel time is 20 minutes (http://www.heathrow.com/airport-guide/getting-around-heathrow/travel-between-terminals). This means you're looking at a possible 35 minutes in travel time alone.
Once you're at Terminal 5, you're still not guaranteed a simple dash to your gate. Terminal 5 consists of three concourses: A, B and C. Only the A gates are located in the main building. If your flight leaves from the B or C concourses, you need to take an additional shuttle train. From there, the actual walk to your gate may take up to an additional 10 minutes.
As a final complication, many BA flights require the use of shuttle buses to get from the gate to the plane. Because of this, BA guidelines indicate you must be at your departure gate at least 20 minutes before your flight departs.
In my opinion, leaving yourself an hour layover an Heathrow is nowhere near enough. I would not be comfortable with anything less than two hours.