If the setting were to change in Hamlet, say from Denmark to England, would anything be lost in translation?

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.