Start by finding out what appears in the title register to your neighbours property. Go here:
http://www.landregistry.gov.uk/
Click 'Find a Property' (at the top of the page).
Enter the house number & postcode.
On the next screen, put a tick against 'title register' and click 'purchase'. (It will cost you £4 but that's a lot cheaper than paying a solicitor to do exactly the same thing!).
It might also be worth forking out £4 more to see what's on the Land Registry's files in relation to your own house. (There would have been no reason for the council to be informed about any covenant between the people who used to own the two properties, so their response suggests that they've already accessed the Land Registry website themselves).
Then, once you're in possession of the full information, you'll know how to proceed. (If there really is a valid covenant in place, requiring you to provide free access along the route now blocked by your fence, you'll have to take it down unless you can get the covenant removed which, in practice, will be close to impossible).
Chris