In most cases the public highway is not 'owned' by anyone (the exception is where land has been bought in recent times for a bypass etc.). To understand that, you have to understand how highways were first created - by people, animals etc. moving across the land to get from village to village. Those claiming ownership adjoining these established routes put up hedges and fences to stop people and animals encroaching - they would have wandered freely across the whole of the established track in the middle ages.
Only rarely can a Council show title to land now forming the highway because these roads pre-date the creation of the County Councils. So Councils are responsible for maintenance but they have no ownership rights - this is why you cannot find any title information.
Legally, highway rights (public rights of way by carriage) can become established once the land has been used in this manner for > 20 years. The verge is part of an original track that became established as a highway but which was never black-topped.
So as a general rule, it is the boundary fence that demarks the boundary between the current highway and the private ownership of adjoining land. However a ditch, if one exists, running alongside the established fence on the highway is actually part of the private land - not part of the highway (don't ask why - its an even longer story than this one).
Getting to the crux of your question, if a verge is fenced off the adjoining owner is guilty of an encroachment. It is not possible to claim adverse possession on the highway after the passage of time of