Historically, control would come from the Foreign Office or Colonial Office with that control moving to the Ministry of Defence in the mid-60's.
I'd look at somewhere like Malta as your primary base, with training & HQ facilities (up to independence in 1964), then moving its HQ & training to mainland Britain (as the Légion étrangère did to France from Algeria in the mid-60's), with bases in Ceylon, Hong Kong, Belize/Guyana, BWI, etc., up to the mid-60's then shrinking to "missions" or "detachments" attached to (but not part of) embassies/consulates.
Force composition/size would probably also follow the French model as the Empire shrinks post-WW2.
Realistically you'd be issuing standard British small-arms & equipment but;
Rifles: My personal choice is the FN SCAR family, preferably -H but -L will do.
Handguns: I'm a .45 fan but a decent .41 (10mm) will do, probably Colt for either.
Machine-guns: You can't go past the ever reliable FN MAG/L7.
Vehicles: Standard British, Landrovers, Stalwarts, etc.
AFV's: you're probably looking at Cav units, not armoured, so British, again, FV100-family (Scimitars, Scorpions, etc.) & FV430-family (Bulldogs, Abbott SPG, etc.), with, maybe, a few non-Brit specialist vehicles thrown in for some variety (LAV-25/LAV-III, etc.)
Air: either/or for AAC/RAF or integral but, if integral, all helo's (Lynx, AH-1, CH-47, etc.) with RAF for heavy lift support.
I'd make the whole of the modern organisation pretty-much air-mobile with Rapid Deployment air & naval forces attached as required for heavy equipment.