Planespeaking:
LHR, LGW and STN have always had a night cap set by the DfT, whilst LTN has a section 106 which limits the number of night movements.
LHR and LGW have always used most of theirs.
In recent years at STN the arrival of Jet2 & increase in Ryanair late night arrivals has used more and more of the spare night capacity available. It results in GA being sidelined.
A similar growth at LTN has seen similar problems develop for GA.
Airlines will apply for season long series of slots and will get priority based on how the slot system works.
As you noted, airports need to be seen as a good neighbour and carefully monitor their slot use so increasingly there won't be much if anything left for GA to use at night.