Only anecdotal I'm afraid, Tush, but then, every case is different. I bought a layby outside my house once, by Adverse Possession. I bought it from the farmer next door, who had used it for many years without owning it.
The Council and Highways Dept had no record of ownership either. It was a simple transaction.
In your case, there is no current "owner" that you know of, so certainly ask the Local/County Council if they have any record. Adverse Possession is a definite possibility providing no one comes out of the woodwork to claim it.
Indemnity Insurance is a very simple alternative. A one-off payment of maybe £100 or thereabouts. Legal advice may tell you that you don't even need it at all.
In another case, I bought a house where the Land Registry's Title Plan showed the adjacent lane in the wrong place... almost in my sitting room. It was quite straightforward to have The Registry amend the plan.
Again, there were no other parties interested, so it was pretty easy.
Only your own local Solicitor can advise safely. As for costs, mostly legal fees. :o(