This article sums up who gets an island quite well.
Basically if it is in a nations 12 mile zone it's theirs automatically and if it's within 200 miles they have a strong claim on it. As shown in the article some naval muscle flexing can extend the zone if noone pushes back. If the island truely is in international waters, whoever plants his flag there and can back up the claim with force gets it. There are no rules as far as I found out, just realpolitics.
Now mind you that these kind of islands are a common occurrence, yet only one in ten survives for a significant time. So governments and even courious explorers aren't fast to claim these lands.
I might even have a good location for your islands. I checked out Google Maps in the satellite mode and along the line you described there seems to be a group of underwater volcanos south of Newfoundland. (I could be wrong, wasn't able to find much else about those) But having one of those rise above the surface seems plausable.
Thus considering the islands position I'm 90% certain that the island would be claimed by the USA. It is the worlds strongest naval power. Portugal, France, Britain, Canada and possibly Denmark and Iceland might claim it, but I only see that happening if 'murica isn' t interested for some reason.
No matter who gets it I dont see how the issue is settled in a non civil way. All these nations are NATO and important trade partners. In fact Denmark and Canada had contested territory in the Arktic until quite recently, yet the "conflict" was based arround flag stealing and leaving whisky and a note claiming the island. No matter the resources, how are they worth a trade war or war with important partners?
Now to the fishes and the jewelery. I don't intend to bash your idea, these are just some points you might wanna consider. Uranium is a quite poisonous heavy metal. In fact, if you where to consume a dose of Uranium salts, whose radiation would kill you, you wouldn't die by radiation. The hevy metal poisoning gets you first. Look up lead poisoning for details. How do the fishes survive the contact with the metal? And how do dead fishes don't poison the entire Atlantic?
Concerning the valuables the quantity of these fishes is important. If they are rare, the resources would not be economically extractable and noone would fight for the island. If they are common they can ruin the market. You see, the high sale value of precious stones comes from their scarcity. (and market speculation and artificial scarcity induced by the traders, but that's a different story) If you now have this island where you can fish up metric tons of these scarce resources they are no longer scarce, so no longer as valuable. The current yearly gem production ranges from 30 kg (diamonds) to ca. 8000 t (amethysts). So if there are lots of your fishes they might upset the markets strongly.
If the island truely is contested viciously a UN mandate or a joint US EU research initiative backed by those countries is consivable. This could be modeld after the Antarctica treaty, as the situation seems similar. A uninhabited island with ample resources. So you might get numerous sientific stations (joined or not) but no resource exploitation.