I don't know about the ownership of the fields above, but the main entrances by the river are on Haddon Estate land, with no vehicular access, as it's a riverside (public) footpath. And it's within the Lathkilldale National Nature Reserve, a SSSI and is also a Scheduled Monument, hence is extremely protected - though this info wasn't very hard to find - well worth a read:
https://historicengland.org.uk/listing/the-list/list-entry/1016755
COPD has a reasonable description of what's in there. There's a simple survey of Mandale Mine in one of Jim Rieuwerts' books, but there's little detail, as there's not much that isn't accessed via the main sough level, and there's not much extra even then. You can find just about everything in one trip in summer, as it's mostly dry - I've been as far as is possible up the sough without breathing apparatus, and it's pretty sketchy up there and a long way from home if anything goes wrong, as there's a roof collapse held up with optimistic recent timbering that must be passed to get to that stage. There's also some flooded stopes, but they empty in summer, so not much diving potential!
It's probably the second-earliest documented mine in Derbyshire after Odin, so it's held in very high regard, and there are no access issues at all as long as you lock the gate after entering - and as long as you don't turn up in a 4 X 4, and
then you might upset someone. Why would you need a 4 X 4 for a trip to Mandale? I find a small caving bag holds more than enough. Unless you're planning to drop the Forefield Shaft instead, in which case you probably would upset someone else, as access is no longer allowed, as far as I know.
I can't work out why you don't know all this already though.