It's not that complicated in the UK. In the vast majority of cases it's either Treasure[0] (which has a special procedure: report it to the local coroner to deal with) or it belongs to the landowner (unless you have an agreement with them otherwise).
There are some corner cases, but you're not going to accidentally find an oil deposit so most of them just won't apply. Detectorists will normally arrange a 50/50 split with the landowner, I understand.
The Treasure Trove Unit at NMS have delegated authority from the Queen’s and Lord Treasurer’s Remembrancer (QLTR) which is the Scots legal institution with, I think, the coolest name.
Huh. TIL! I knew Scotland doesn't have a coronial system, but I assumed given the relatively recent reforms to treasure in E&W similar rules would have been brought in UK-wide and cases would just go to the local Sheriff.
Looking at the Scottish Treasure Trove website Scots law does look more sensible than the old English position, so I suppose they just decided not to interfere.
One of my hobbies is scanning comments on HN and when someone mentions something about a "UK" legal or education matter jumping in and adding a "Not in Scotland" comment - I really should automate the process ;-)
Fossils are not artifacts, they weren't dropped or cached by anyone. They have never been anyone's property. They are "natural". The rules are therefore different, more akin to someone picking mushrooms than someone finding a lost gold ring. When something has never had a human owner, gaining first possession can really matter.
Fungi, flowers, fruit, and foliage (growing wild, not cultivated) are one of the corner cases - they're specifically carved out by s.4(3) of the Theft Act 1968. There are a few others, as I said: mineral rights, wild and semi-wild animals, etc. But otherwise it's not generally complicated (at least in England and Wales).
Now, you have a good case that the natural things shouldn't belong to the landowner. But English law has chosen not to agree with you (unsurprisingly given the strength of the landed interest over British history). First possession is not the rule here, it's the ownership of the land it's found on that matters.
There are some corner cases, but you're not going to accidentally find an oil deposit so most of them just won't apply. Detectorists will normally arrange a 50/50 split with the landowner, I understand.
[0]: https://finds.org.uk/treasure/advice/summary