Sounds strange that they can claim this freehold when there is nothing on the deeds. It's the sort of thing that would/should have come out in the searches when you bought your place.
Your neighbour - more importantly her partner - probably knows that she's on very shaky ground; hence the...
And charge you accordingly - it's way cheaper to ask NSC!
Anyway, I'm intrigued about the dispute in the first place. You never know; you may get some decent advice from someone on here who's had a similar dispute.