I wonder if BA practise engine failures in the simulator using the autopilot? It would seem sensible to use it as you can on the A319, as long as the rudder is trimmed, but simulator base checks normally require manually flown engine failure procedures.
Yes this is practised. Engine failures with and without autopilot. Of course the engine failure on takeoff won't involve an AP but at 100ft this can be engaged all the way down to a CAT3A autoland which is recommended to reduce workload.