In the UK, ALL treasure belongs to The Crown unless otherwise determined...............
https://en.wikipedia.org/wiki/Treasure_Act_1996
It’s interesting how the law works. If something is lost, it’s pretty much finders keepers. But if something was hidden, it belongs to the Crown. How is lost/hidden determined—by the coroner. A coin here or there is usually considered lost, while something deliberately buried, like a hoard of coins was hidden.
As noted above, the Crown wisely pays fair value for everything local or national museums consider historic. This also allows ‘experts’ to do the recovery and record the history of the site.
Treasure in the water gets really complicated. If a sunken ship was a national ship, it and everything aboard still belongs to its home nation. A treasure finder might well end up with nothing for his troubles finding it. That has happened with Spanish treasure ships. The laws covering these kinds of things are often very old and certainly don’t seem fair.