Keeping part of the find is probably like the guy in the US who found a way to shelter the few 1933 $20 from the melt pot. He kept them stashed away 70+ years only to die and have his heirs get advised to go to "the govt" to have authenticity evaluated. Bet after legal fees they took a bath.
U.K. museums and research institutions get first preference for the finds if they can pay. Then other museums if they can pay. Price is set at fair market by an independent board. Any that aren’t bought by a museum belong to the finder and landowner. There probably are some cases where the finder and landowner did get to keep some. I’m not familiar enough with the various finds to cite any where museums didn’t buy them all. It’s also possible that some coins bought by museums were later sold to the public by the museums. Wikipedia (https://en.wikipedia.org/wiki/Treasure_Act_1996) has a good description. BTW, the current treasure law only covers coins made of precious metal. Those of base metal belong to the finder and landowner unless they’re part of the same find as precious metal coins. The law is being considered for revision to cover coins of all metals. Cal
Calcol got to this before I did. If no museum wishes to purchase the coins they go back to the finder and landowner. I would also suspect that if they are purchased the museums do sell of some of the duplicates. This find will definitely be declared treasure due to the high quality of the coins, the mule pieces and the previously unknown mints.