Max,
Would suggest something like:
Biggin - M23/M25 VRP
M23/M25 VRP - Farnborough
Farnborough - CPT (With a little side-step to avoid Blackbushe)
CPT - Welshpool (Pretty much 315ish true track) will take you just slightly west of the brize O/H. You just need to watch out for R101, R104, P106 and the HIRTAs at Pershore & Defford.
Give Brize Zone a call on 119.0 in plenty of time (pref as soon as your DME picks up their TACAN (111.9)), they'll give you a squawk and may give you a height but you'll have no probs
Oh, one other thing that may catch you out if your're not used to dealing with the RAF: Some mil controllers will say "maintain Victor Mike"! It's nothing sinister just their way of saying 'Maintain VMC'. . . .
Have fun!
BH