We flew British Airways from FCO to LHR in January 2013, and things may have changed since then, so as suggested earlier, check the departures board for your terminal. I don't know whether BA is now using self check-in kiosks at FCO, but otherwise check your luggage and obtain a boarding pass at the BA check-in desk at FCO. The security maze had a seemingly very long line at FCO, but it moved very fast and we were on our way to our gate in short order. Approaching the gate, a line had formed early on, with everybody standing and waiting for the longest time to be let thru a door. We then all reached a second door, where the line continued to stand until it was time to actually board the plane, so sitting much beforehand may not be an option, although your results may be different now.
T5 at LHR is big and busy, but it works. If you find yourself in T5, get a bite to eat at the "Eat" stand or at Gordon Ramsay's "Plane Food" restaurant. Get some Thornton's toffee at The Chocolate Box shop. If you need assistance at LHR or have questions in any terminal, ask any of the staff walking around in a purple coat. Have a great trip!