They have the right to exercise extraterritoriality, i.e. exercise their own legal jurisdiction etc. as do embassies etc. however this is at the gift of the host nation and isn't automatically conveyed by the act of leasing land.
The exact details governing Fairford, Mildenhall etc. is probably down to an inter-governmental agreement but it doesn't stop it being UK territory, and if we thought they were up to something dodgy we could probably remove the right. Certainly this is the situation with embassies as I understand it, i.e. if they are conducting activities not in keeping with their status as an embassy then it stops having diplomatic immunity.