You are right about the night quota system - the DfT sets seasonal limits on movements and 'noise points' for LHR, LGW and STN between the hours 2330-0600 LT.
Noise - or QC - points are based on the a/c certification and are either 0.5, 1, 2, 4, 8 or 16, althout 8s and 16s are banned. The QC doubles for each 3 dB extra noise. NB: the max is 16 - otherwise Concorde would have been about 4096 points per movement!!!
Each airline gets an allocation for planned night flights (typically long-haul arrivals at LHR) and must manage within their allocation. There's a small pool for unplanned late-runners by airlines without a quota allocation, but its strictly limited and can run out so you may see the odd divert to LTN or somewhere.
The earliest scheduled flight at LHR is the BA016 SYD/SIN at 0445.