@landmark orbiter ,irrespective of whether it is a public footpath ,or more commonly known as a public right of way ,if it is still on the NT owned land ,then any TOAL restrictions that are in place still apply
when it comes to NT, or any other organisations that govern land use, then although they cannot prevent you from flying over their land ,unless there is a specific FRZ in place ,such as the one at Windsor Castle ,and many other places ,they can still if they want to ,take out a civil case against you for trespass, or nuisance ,and they have much deeper pockets than most members of the general public do
where I live in Wales we have the Brecon Beacons National Park a huge swath of land, that covers a great area, they do not allow drones to TOAL on their land , and rigerously enforce this ,with rangers, who patrol the Park, and can appear from noware any time ,had a run in with one of them recently, flying in an area of the park ,and his main arguement, for me not being allowed to fly there, was good old TOAL,after i had landed, we had a conversation in a calm manner ,he putting his points accross , and me mine on the subject ,the outcome was that he aggreed that the park did not control the Airspace ,and as long as i had the particular land owners permission
where i wished to fly ,then there was nothing he could do to prevent me from flying ,as long as i followd the drone code,for the drone i was flying , in this case a Mini
3pro, and didnt endanger people or wildlife ,so as long as you ask the land owners permission, where the foot path crosses their land ,even if it is in a NT area ,you should be good to go,just check for any airspace restictions ,before you fly