The issue of whether or not planning permission is required is potentially a red herring and ultimately probably won't be of any real assistance to the OP.
Should the driveway require express permission, either because it fails to comply with the new Class F rights of the GPDO in terms of construction/materials or because it's not within the curtilage of the property it serves, then the neighbour is still entitled to make a planning application to the local planning authority for its consideration.
Whilst the issue of whose land it actually is may well crop up, if the LPA is satisfied that the certification relating to ownership is correct, then it may well grant permission anyway.
Even if the neighbour did concede it was not entirely on their land and served the appropriate notice on the OP, planning permission could still be granted anyway.
In addition, it's important to bear in mind that even if the driveway does require planning permission, the LPA would have to consider whether it would be expedient for them to pursue it in the event the neighbour failed to submit an application for their consideration.
Unless its construction is likely to turn the road into the Ganges when a bit of rain falls, they may well conclude that are no justifiable reasons for taking action in what is essentially a private neighbour dispute.