In reply to RyanC:
I imagine bus is the only public transport option between the two, there isn't a train. Brean Down and Uphill Quarry are probably the two other closest crags to Burnham. Again, I would think bus is the most likely option.
Here you go:
http://www.firstgroup.com/ukbus/bristol_bath/journey_planning/maps/weston.p...
Looks like there is no direct link between Burnham and Cheddar (at least with First Bus), you have to go via Weston. Brean and Uphill look easier.