Tim,
There's a group called
Tiger Fly at Wycombe who do flights in a Tiger Moth and, I think, a couple of other types. I'm not sure if you actually get to handle the controls or not, and they're pretty pricey.
The flying club at Cambridge still use Tiger Moths, and I flew a Tiger Moth with the Tiger Club (at Headcorn), although both are a bit far from you.
I think that the club at White Waltham have a Piper Cub: I'm not sure what 'flavour', but FFF can probably tell you.
Otherwise it might be worth having a look in the "where to fly" guides, as these may have ads for "interesting" flying.
Brooklands