Ignoring the ...ahem... unfortunate Phantom vs Jag incident (and the alleged shooting down of an abandoned Harrier GR 3 heading sans pilot towards E. Germany by Lightnings, etc) and sticking to conflicts, the last confirmed air-air kill scored by an RAF pilot in an RAF-owned (not FAA) aircraft would have been in World War 2.
[Edited to add - No it wasn't, you idiot! In fact, it was probably the clash between RAF and Egyptian aircraft in 1948 when the EAF lost a couple of Spitfires to RAF Spitfire FR 18s....]
Unofficially, though, there are strong rumours that there was at least one air-air in the Indonesian confrontation when a Javelin brought down a C-130. ISTR that Jackonicko said 'the evidence is there if you know where to look', but I've not looked there myself yet. Perhaps I will...
There are a couple of less well-substantiated claims about a Hunter and a MiG (-17?).
If the ownership of the airframes is removed from the equation, then the Falklands was the last occasion with the RAF providing about 25% of SHAR pilots who scored about 25% of the victories. The joint leading scorer was from the RAF (Flt Lt Dave Morgan), although he subsequently transferred to the RN.
RAF pilots also scored victories flying with the USAF during the Korean war, and, IIRC, there was one scored while with 77 Sqn RAAF on Meteor 8s. However, IMayNot RC there...