Not able to download CAP682 to have a proper read. However AFAIU :
Flying training for standard fixed wing (e.g. not microlight), has to be conducted at a licenced airfield. There needs to be someone observing the manouvering area and runways.
Fixed wing flying training can happen at any licenced airfield, whether it be A/G, FIS or full ATC
In fact I'm not aware that even an A/G service has to be available - e.g. some airfields, such as Sleap, have an A/G frequency assigned but carry out training when it isn't manned. The requirement to have someone observing could, I guess, be undertaken by the fire crew.