We had shared access at our last house and, although we owned the access path, it still caused problems. The house next door to us was rented and we had to explain to every new tenant that we owned the path, that they could use it but couldn't block it etc etc etc. We've learnt from that to avoid properties with any form of shared access because it just brings hassle.
Our understanding also was that you couldn't put a lockable gate on the path even if we gave the neighbours a key. Your neighbour certainly shouldn't be putting a gate up on your land so I would get them to remove it for that reason alone.
As has been said, show them the Title Deeds and that should, hopefully, put an end to it. Unfortunately, some people are just either stubborn or a bit thick and it's possible you might have to get a solicitor involved but I'd save that as a last resort.