Things may have changed recently but as I understand it the MOD mandate is for RAF stations to always make their facitities available for revenue generating activities unless it seriously impeded operational flying.
I fly into RAF airfields frequently and once you have the paper work organised and accept it is both their ball and their garden the whole process is fairly painless. And as for diverting they are fantastic. Once asked to divert to Brize when 1 mile final on a training ILS (we had only just broken cloud so was obvious we wouldn't be able to get back to home base). We got the affirmative decision in about .5 of a nano second so landed. Free coffee and biscuits and no landing fee. Sat on the ground until the front went through and then recovered to base .
So just ask
PS there is an amazing amount of aerial activity at both Benson and Odiham at the weekends. If you can't see it on the ground I'd look around cos it's probably up there with you