There are four tourist visitable "caverns" (a large amount of the underground space in them was actually created by mining) around Castleton, in the dark peak.
- Treak Cliff Cavern
- Blue John Cavern
- Peak Cavern
- Speedwell Cavern
Of those, I would specifically avoid Speedwell - the tour is on a boat, and there will be no way out. I was in Treak Cliff cavern last week, and while the tour is guided (as it is in all of the Castleton mines), the whole route was only a few hundred metres, and the exit fairly clear (we were directed to find our own way out when we wanted at the end of the tour) - I'd imagine with a quick word to the guide, you'd be able to leave alone. The others it's been years since I visited, but I'd imagine a quick email to the operators would sort things out.
A bonus for Castelton is that it is a very touristy village. If you have to leave the caverns, you can go: Walk up Mam Tor/The broken road/Cave Dale; shop for unique Blue John jewlery and trinkets; eat/drink at the many pubs/cafés; visit the local castle ruins; escape to Hope or Edale and explore some peak district wild lands.
The visit peak district website provides information on these and other caves. I won't comment further having no personal expereince of any however.