There are at least two scheduled BA flights which arrive to LHR before 0600 in the morning. Both flights are from HKG, BA026 STA 0450 and BA032 STA 0500.
A year ago I was actually on board of BA032 and we landed exactly 0500 in the morning. It was quite a hilarious experience as there was nobody to drive the airbridge or connect the external power after we parked to our stand. The APU seemed to be faulty and after the captain decided to turn off the engines we lost all the lights in the cabin except the emergency lights of course. After waiting of 10mins some sleepy gate agent arrived and attached the airbridge and we were able to disembark in almost total darkeness. Luckily the F/As helped us with flashlights...
Best Regards,
AM