The process of switching to your BA flight will be more unpleasant than you would expect reading that a bus is all you have to take. Here is the process:

After stepping off your incoming AA flight, follow the signs to Flight Transfers, a fair distance away.

Carefully watch the transfer signs, as the queue and process is different for passengers transferring into the same terminal than for passengers moving to a different terminal, as you will. You will eventually go down an escalator to a bus lobby where you wait for the transfer bus. Be sure you head to the appropriate door for your target terminal (1).

Cram onto the bus and be transported around the airport. Between the wait for the bus and the actual drive time, plan on 15-20 minutes of delay from when you reach the bus lobby.

At the new terminal bus lobby, head up the escalators where you will be routed to a security screening area. While you don't have to clear customs or deal with passport control stations, you will have to go through the metal detectors and run your carry-on items through x-ray systems. This is where it can be ugly -- at peak times the queue wraps around inside a large hall as much as for a top Disney amusement ride, then snakes outside and back up the hallways towards the bus lobby. It can burn up quite a bit of time as you inch along with many hundreds of other passengers.

Once you thread through the eye of the needle -- the few x-ray and metal detector stations -- you then are in a lobby with flight service desks for the airlines in this terminal. If you need to change boarding passes, attempt to hop on an earlier flight, or are not already in possession of a valid onward boarding pass, then you would queue up here for the BA counter. If you have thru boarding passes, just ignore these and continue to the escalators.

If you are flying in a premium cabin on BA, sometimes there is a fast track lane open in the large hall, permitting you to skip over a few hundred people for the x-ray and metal detector station. Sometimes it is closed, however, leaving premium passengers to queue with the masses. This special track generally does NOT accept AA status such as EXP as a substitute for a premium cabin boarding pass on BA, so if you are in economy on BA then plan on reading a good book while standing in the lines. When you are ticketed in business or first but the fast track lane is closed, read a good book while inching along.

Head up the escalators and watch for your terminal -- 1 -- to be sure you get to the correct floor.

If you are a PLT or EXP then head for the BA lounge, anyone can head there if flying in J, otherwise find a plastic seat somewhere. There is a fair collection of shops to wander through. The monitors will specify your assigned gate for the flight to FCO when close to departure, before then you just wait, either outside or in the BA lounge. Once the gate is assigned, you can plan for when to head down -- some gates are a bit of a walk away, others are closer and need less lead time.

The minimum connect times used for booking these AA to BA connections are very tight in most cases, with all the opportunities for waiting and delays, but most times you will make it unless your incoming flight is late arriving. If so, you will worry until you get through those metal detectors, at which point it is an easy lope to the boarding gates even if you have only 20 minutes or so left.

Have an enjoyable trip