I doubt any local folk (self included) will be heading underground.
However just to tickle your tastebuds within about a 5-15 min drive you have:
Sanford Levvy.
Singing River Mine.
Stainsbury's Shaft.
Star Shaft.
And on the top of the Mendip has more. Most of the options mentioned above are on the C.S.C.C. key, (available from caving huts for a non member deposit, (or your local fiendly Mine Explorer in return for tales of daring do, and beer, Nudge, nudge!))
I'm kinda busy this weekend, decimating the back garden. But, should you be "innarea anytime soon" Drop me a Pm and perhaps we'll be able to get some trips in!
Better to regret something you have done - than to regret something you have not done.