Hitler was certainly not a monarchist. The characteristics of fascism are always national and often contradictionary, and always nationalist. Hitler did not want a return of the German Empire, and wanted to keep ultimate power with himself, not like Mussolini who technically was appointed by the italian King, who deposed Mussolini when Italy was invaded by the Allies. Oswald Mosley used the imagery and appeal of the monarchy to gain legitimacy, much like Franco in Spain or Hórthy in Hungary (though neither were "true" fascists) were regents in monarchies with no kings, so they could keep power.

The nazis had ideas of Germany and England allying to secure global dominance, with England ruling the seas and Germany on land, and in that case the Royals would very likely have been the figureheads of Germany. If Germany somehow managed to invade England and capture the King, who knows? A propaganda pawn, execution, puppet leader or exile in the colonies - the specifics of what would happen to the Royals would have been decided by political circumstance.