Well, let's get down to the nitty gritty. The first document in the link above is a pdf called "Summary of Existing Access in England" and does not mention caving at all. The second called "Outdoors For All - Recommendations to Government on Access to Nature" mentions caving once, as follows:
"2. Extend statutory access rights for a broader range of recreational activities, including but not limited to wild camping, paddling, swimming, cycling, caving and horse riding.
As well as expanding the types of landscape to which the public has freedom to roam away from public rights of way, the majority of submissions argued in favour of expanding the type of activities allowed. Cycling, horse riding, paddling, swimming, caving and wild camping do not currently enjoy rights of access, limiting opportunities for public enjoyment of the outdoors. Several respondents noted that the Secretary of State could amend Schedule 2 of the CRoW Act. 'Higher rights’ like wild camping, horse-riding and swimming are explicitly excluded on CRoWAccess Land under Schedule 2 of the Countryside and Rights of Way Act (2000)”
This is very unhelpful of the committee and it's nothing to celebrate, i.e. them saying above that caving does not have a right of access. Well, I'm afraid that actioning their recommendation for fixing that issue (as they understand it) will not help the caving situation because caving is NOT prohibited under Schedule 2 of the Act unlike cycling etc on Access Land. So there is no prohibition of caving that is amenable to removal by future legislation. The right already exists as expressed in the first few pages of the main Act, which has led to the argument with DEFRA etc about what the terms "open-air recreation", "access land", and so forth embrace.
DEFRA (and NRW in Wales) are not going to change their minds, and the legal action by way of BCA's judicial review to have the Courts change their minds for them was abandoned on the steps of the courthouse due to lack of funds and strategic problems with the way BCA's case was managed that need not detain us here, so we will have to live with the present situation.
My suggestion is that we just ignore DEFRA and NRW, keep calm and carry on. Our opponents will definitely not ever bring a legal case against any caver or caving organisation. They fear the courts, as the JR case amply demonstrated, because the quangos know their argument against access rights to go caving are contrived and threadbare and they would be exposed as such in public by airing them in Court