Complicated garden dilemma

Land_Registry

Couple of things to add to what’s already been discussed/shared 
Firstly if you have a mortgage then what you want to do may not be the key as the lender may insist on a certain course of action
Secondly if you confirm the purchase but ask the seller to register the possessory claim first, then they can ask for the application to be expedited. Expedition reduces the wait time from it being expedited to being considered to 2 weeks. 

However the downside is that it may still take 6/7 weeks to complete as a site visit/survey would be needed (if the stat decs were ok) and then wider checks completed with adjoining property owners in case the lands is there’s for example
So plenty to consider including whether anything you’ve suggested is actually OTT as well as timescales plus Practice Guide 5 from a registration perspective re the claim 

StarvingArtist

Thanks for replying to my post. There is a lot for us to think about. We have no mortgage so it's our decision.  At the moment the expedited timescale seems a bit too long to wait, but we have yet to find out what the seller thinks about it all and will add that to our considerations. I have found the Practice Guide which is very detailed and helpful.

robatwork

The vendor and EA have an axe to grind so can't be relied on to tell the whole truth and nothing but the truth. 

If it was me I'd be knocking on both adjoining neigbours' doors, introducing myself as your new neighbour, and seeing if they either have been granted adverse possession or at least that they have had unfettered access to that area of their gardens for x years.