In England, I believe that the British Historical Trust (or similar) gets “first dibs” on finds, based on an independently (and that’s key) calculated market value. If they pass, you can sell it. At least that’s how it worked a few years ago...
If one of the museums or trusts wants it, they have to pay the finder the full market value, as determined by the ‘independent expert’. If the government organizations don’t want it, it probably isn’t worth much.