Just one point from the
Aircraft Noise Enquiries & Complaints Policy document at
HOW TO MAKE A NOISE ENQUIRY OR COMPLAINT
2.2 Abusive or threatening correspondence
Our staff will not respond to any correspondence that could be considered of an abusive, discourteous, or threatening nature. Where we deem it necessary, such complaints will be handed to the police for investigation. It is also our policy not to respond to a complaint where we deem there to be an misuse of our complaints service, for example by use of alias names.
Sections 6.0 & 7.0 seem to also apply but as the document is dated Aug 2022 it might have been recently amended.
A check on how LHR does things shows similar procedures "such complaints will be handed to the police for investigation."
The whole document is geared around flights which have deviated from agreed flight paths or outside agreed hours so his grievance that everything goes over his house maybe party valid if he happens to live directly under a flight path agreed with CAA & NATS. If an aircraft is going outside an agreed flightpath without good reason LLACC can action. If it is not then this is outside the scope of this complaint procedure.