Further to my last, I thought I should just comment upon the treatment you said you received on your flight from EBKT to EGLM.
If you look at a map, you will see that airfields such as EGLM, EGLK, EGTF, EGLF are badly placed for IFR arrivals from the East. The problem is that all levels up to 6000' are occupied with traffic departing LL and KK on SIDS. To stop those SIDS off at, say 4000' from LL to allow you to descend to 5000' on your way into LM will mean certain KK SIDS might have to be stopped at 3000'. The noise complaints and pilots' complaints would rocket. Therefore the only way in is to remain high level (FL 80+) until you are clear of the SID tracks, which would mean going almost to CPT before descending. LF inbounds from the East have to go CPT-HANKY-PEPIS (officially) for just this reason. There is also a procedure by CODEY but this is seldom used.
In addition to all that, the density of arriving traffic for LL can mean that blocking FL80 (say) at BIG and OCK is a big problem for the LL golden boys to cope with with their vectoring for approach.
Because of this, you will usually be offered the choice of a high-level transit and then a dirty dive for your destination (shockcooling a possibility
) or descending to leave CAS around DET and then routing with Thames Radar and/or Farnborough Radar. This is particularly the case for EGTF but is also valid for EGLM and others.
If you are departing to the east, you might consider that you will get a more direct route and an easier clearance if you stay underneath the TMA and join at DET or BPK/LAM as appropriate.
I hope this helps a bit.