"The Title Register for the area has belonged to the same person since 1984..."
In my experience, the title register shows the address of the person who owns any bit of registered land.
For example, last year I was interested to find out who owned a piece of land. It was just described as "land at Holkham, Wells-next-the Sea".
There was no address for the land, it was just a piece of land so I had to do a map search.
After I identified the particular bit of land that gave the Inspire ID which can then be used to find the Title number of the plot of land. This will then give you the name and address.
Did the title register really not show the address of the registered owner?
[EDIT]
Or are you saying that the owner no longer lives at the address on the title register?
So does it say something like the owner is Joe Bloggs, 24 End Terrace and there is no longer a Joe Bloggs living at that address and nobody remembers him from back then?
In that situation the ethical thing to do would be to try and track down Joe Bloggs (or his descendants) and offer to buy the land from him/his descendants.
Alternatively, you can go through the process of adverse possession. When you make a claim the Land Registry will write to the registered owner and they then have 65 working days (typically three months) in order to object and give a counter notice.
If the registered owner does not object within those three months then you can apply to be the registered owner of the land.