It might be a bit after the period you are looking at but Peter the Great was fond of disguising himself in public. Peter the Great: His Life and World is an interesting read. Many stories like the one in question.

A followup question:

Would they have been required to change their appearance other than just wardrobe? Could a regular person recognize the King/Queen's face during this time period?

Kind of an add on question if there aren't any know occurrences of it. Did this myth start because of Shakespeare's Henry V, or did he draw on earlier stories?

Huh. The Elizabeth II fact feels like something I vaguely knew about but with no specifics. Anyone have more detail?