It is pretty simple really: land is 3D in law. Whoever owns the land owns the access rights to the material the land is made of and the air space above it, to the centre of the earth and the end of the universe, only in theory, which is the medieval dictum: Cuius est solum, eius est usque ad coelum et ad inferos.
These days the 3D aspect is limited to only a reasonable depth or height. It is not all the way to heaven and hell as in the latin dictum. For example, you personally cannot charge an airline a trespass fee for flying through your airspace which happens to be a commercial route otherwise airports and flightpaths would be impractical. Monumental numbers of wayleave agreements with homeowners and commercial property would be needed. Similarly, modern case law suggest rights going downwards are limited to what you can in practical terms reasonably exploit for yourself.
Mineral rights can be sold off separately from land which splits off a right to exploit the minerals commercially as distinct say from farming the land surface commercially. The existence of mineral rights, whether explicitly created by their sale or remaining integral with the land if not sold off, is irrelevant to public access rights.
For example, limestone pavements at Malham are bare minerals appearing at the surface. Someone owns that land; someone owns its mineral rights; they would be the same person or entity unless the mineral rights had been sold. You can walk on that grass or rock because (a) it?s CROW Access Land, (b) there are footpaths and other rights of way. You can go rock climbing legally too where you are on the minerals until you fall off and then you are in the landowner's airspace for a few moments and finally land on their grass or fall down their cave.
On the other hand, public access may be limited by by-laws which can apply to NT land or NNRs. CROW and LPA both provide the means of limiting public access too, for reasons such as public safety or conservation in specific cases. SSSI status of itself is irrelevant since WCA offers no means to limit public access to SSSI land.
There is no difference between walking on bare limestone above Malham Cove and walking on bare limestone anywhere else - like in cave. We?ve just had a Judicial Review where the Welsh Government and its allies have walked over burning red hot coals for 2 years to try to stop a court case that would have determined the principle that an access right to land means an access right to caves too.
My view is the government can never bring a court case against cavers on this matter, and they would do their utmost to stop anyone else trying it by joining their case as an interested party and then frustrating it. The government clearly does not want an explicit and crystal clear right of access to caves to come into existence thus it simply can?t handle the uncertainty of what any judge might rule. It's paralysed and so it's fallback position is relying on an argument that the general grant of access in CROW is not general enough to cover caves. It's saying an unspecific term was cunningly crafted so as to deliberately exclude caving so as not to have to bother adding caving to the explicit list of banned activities in Schedule 2 of the Act. This is close enough to nonsense to fear litigation on it.
Mineral rights simply don?t come into it. They concern commercial exploitation. You can walk on the minerals of the limestone pavements above DYO on Access Land as of right (these are on Welsh Water land) and you can walk on limestone pavements on the land belonging to the farm too as of right where it is Access Land. This is regardless of the fact that the farm sold its mineral rights to DYO, and regardless of mineral right ownership in Welsh Water's area, and regardless of whether you are on the minerals or on the grass be that for walking, running, climbing, caving or anything else that is not on the banned list.