The answer to the question is: Yes, the B777 does compensate for yaw automatically during a power loss. Actually, there need only be a 10% power loss between the left and right engine to affect thrust assymetry compensation.
The amount of rudder supplied is about 90%, leaving some room for the flight crew to 'fine tune' the final trimming.
I'm not sure of BA's SOP regarding their engine failure procedure, thus can't answer for a possible thrust reduction.
If the aircraft was flown in VNAV, there are several automatic functions that take place during the flight profile and it wouldn't be fair to make any guesses of the flight crew as to what was happening on their aircraft. Suffice to say, the airplane is a breeze to fly when all the automation is working correctly.
"Engine Failure?......Ho Hum...ring the bell and ask for another cup of coffee while this thing sorts itself out"!