If the setting were to change in Hamlet, say from Denmark to England, would anything be lost in translation?
In terms of the words, somewhat. Going from one language to another will have an impact on the work. The impact may sometimes be small, but there is a chance that the original meeting and intent would be lost or altered by the changes. In terms of the actual story, I would have to say no. Many productions and films of Hamlet have changed the plot's setting from Denmark to elsewhere yet the overall plot has not been altered nor has the impact been reduced.
This comes from a fan of Shakespeare, and I admit my bias may be showing. There are purists, however, who would argue that changing anything in the work would cheapen it, but I disagree. If a work is strong enough, if the characters hold the audience's attention, then where the story takes place does not matter. Hamlet could take place on Mars and the story would still have the same effect as it has had for centuries.
I don't think Hamlet is a particularly Danish play. Hamlet has a plot where the king is a murderer and his wife is a bad woman. This is controversial stuff! Shakespeare lived in protestant Elizabethan England. It was a very unsettled time and there were lots of plots and spies and plans to replace Elizabeth with a Catholic monarch. The country had been through some tough times as protestants and catholics both tried to take control of the throne. Elizabeth had spent most of her reign trying to discover Catholics plots to overthrow her. Trouble makers were executed without much trial or hesitation.
Playwrites could get in trouble for writing controversial topics, so Shakespeare uproots his potentially threatening story and moves it to Denmark to make sure people didn't look for 'messages' or 'parallels' about the English monarchy. But I don't think there is anything that makes it Danish. Shakespeare never went to Denmark and we know of nothing that connects him to it.
If you relocated Hamlet to France or England or Italy, I don't think you would lose anything. It may be nominally set in Denmark, but I don't think its location is relevant.
Compare it to Venice in the Merchant of Venice or Othello. These stories need Venice as their setting because they reflect lots of things about Venice and 16th century Venetian society. But Hamlet is not especially Danish.
Denmark's ambience of a dark, backward inbred village works better with witches, ghosts, graves and insanity than a sense of a more educated population and more practical minded English population. The Gothic setting of Macbeth's 'castle' deep in the woods where owls shriek anddarkness falls like a black evening cloak seems to make prophecy, murder intrigue and madness more at home than in an ertstwhile English setting. After all, Hamlet is in a sense a Gothic drama - an inocent girl, a dark secret, an isolated house in an unpopulated area and a complex protagonist we don't know what to make of. Yes, Denmark is the place for the plot to wind and unwind.