Depending where you are in the south of England, the caves of South Wales aren't too far away. Maybe consider Eglws Faen (spelling may not be entirely right!) near Crickhowell. Give some thought to where you park before setting off walking; I think there is a history of grumpy locals being less than pleased with folk who park inconsiderately.
This cave is pretty easy and has a biggish chamber near the entrance from memory. Daylight may come in the first bit but you always have the option of a night trip. I seem to remember doing a through trip from here to another entrance using cigarette lighters but I ought to stress I was with a bunch of very experienced cavers and I'm certainly not recommending this. In fact it'd perhaps make sense to contact an experienced local caver wherever you end up filming if you're not a caver yourself; there are hazards in all caves, which you may not immediately recognise.
There is also Powell's Cave alongside the road up to the South Wales Caving Club HQ, a big and easy through trip. In a certain northern caving club's library there is a photograph taken just after the war of someone riding a motorbike through it, complete with goggles and greatcoat. (This kind of stunt would be very much frowned upon nowadays, in the interests of conservation. Which leads me on nicely to mention that you also need to make sure that nothing you do damages the cave environment in any way.)
Whichever part of the country you end up in, maybe post on here when you've chosen the venue and ask if a local caver is willing to meet up for a couple of hours, just to oversee what you want to do and make sure it has a happy ending - for you and for the cave.