Probably ILS calibration flights.. You have to do them once in a while, better to do them when it is quiet, like around 1 'clock at night...
I concur with 'the_stranger', most likely a calibration flight. They are often done with Beech KingAir/QueenAirs in Europe, so maybe this one is of a similar size and type.
On this subject, one has noticed, through a bout of insomnia, that there seems to be an aircraft, a guess at an Airbus A319/20/21 doing circuits somewhere between 01.00 and 02.00hrs into LHR. My thinking is that it would be a training flight/test flight of either BA or BMI. Any ideas ?
Regards
SHJ