Sad to say, but there are no caves of any significance in the Mendip Hills. For proper caves you need to go to Devon or the Forest of Dean, neither of which are easy to reach unless you own a car, pony & trap or pogo stick. If you are lucky enough to own all three, then you're too rich to take up caving anyways and you should take up golf instead. Most golf courses have more holes than any of the caving regions in Britain.
Good luck on your quest, Badgersarse!