The Mendips are predominantly slippery limestone and clay.
Theres loads of good, steep, rocky riding around Cheddar, all of which is pretty unpleasant and death-trappy at the moment.
Per Radbikebro, I’d head for Rowberrow, but be prepared for your standard, plantation style man made stuff. Some will say its a bit ‘natural’ because it doesnt have a load in the way of aggregate surfacing, but theres not so much natural about riding in a monoculture on planned/built trails with cookie cutter lines IMO. YMMV of course. It’ll be good for a few hours either way.