Having been in the "Treasure Game" for several years, I believe that England does have some sort of compensation program. Others were allowed to keep a small percentage of the find, but, not enough to reduce the "supposed" value of the bounty.
Any land finds, unless on forbidden to hunt areas, are deemed the property of the finder, in MOST states, in the USA. ALL National battlefield items from Civil War, Little Big Horn, etc. are off limits. Old Fort sites are up to discretion of the land owner to give permission to hunt.
The biggest thing will be to do restoration of the coins using reverse electrolysis. The coins are coated with corrosion, as oxides form over the years. The reverse electrolysis changes those oxides back to hard metal, for the most part. It will stabilize the coins so further deterioration is not a problem.
When we did the shipwreck artifacts, we made up small racks with SS wires and clips, to suspend the iron or silver items. Gold and Bronze-brass does not need this, just a good gentle cleaning will do.
When Treasure from shipwrecks is found, in Florida, the State gets first choice of 20% of the total estimated value as salvage. Any Particularly fine or unusual item, USUALLY goes to the state, unless a significant amount of booty was found. Then, it is possible to negotiate with the State.
When we contracted with Mel Fisher, the State got 20%, Mels company and us split the balance 50-50. They would do the restoration in their labs, catalog the items, and once a year there would be a settling up.
Usually, extra fine items would disappear or just not be able to be located, so, a pittance was offered instead.
We never "found" anything of great value, although some salvors did. We found coins pieces of gold washed silver statues, cannons, and the like.
My youngest daughter worked first as a lab assistant, then became Mels Technical artist, doing drawings of every elaborate item brought up from wrecks, then eventually became Lab Curator. Internal BS caused her to quit, so, others got to brown-nose Mels daughter.
I have a few photos, but need to take Digital images to post them, if anyone is interested.