Are you absolutely sure that the land is unregistered?
Apologies if you've already done this.
For others reading this, unregistered land is not very common at all today but does still exist.
Perhaps the easiest way of finding out if a piece of land is registered or not is to have a look on the Land Registry map here:
https://www.landregistry-uk.com/map-search
This shows all registered land in England (there's a slightly different process for Wales).
If the bit of land you're looking for has a box and a number covering it then it has been registered. If it's just blank and there's no line surrounding it then it hasn't been registered.
Attached is a portion of the map looking at Woodley near Romsey. You will see that there are quite a few houses that are still unregistered.
If a piece of land is registered then you can take the number shown on that map and go to:
https://search-property-information.service.gov.uk/search/search-by-inspire-id
It will then cost you £3 to order the title deeds and that will give you details of who owns that particular bit of land (the land registry charge £12.95 if you do it directly from their site).
If a piece of land is shown as unregistered on the map and you have possession of it then it is a straightforward matter to apply for adverse possession if you have the necessary documentary evidence.
As to cost, that will vary. The easiest thing would be to just phone up a local solicitor who does conveyancing and speak to them. Depending on how much they charge you can then decide to go ahead and use them or do it yourself