In British English they'd both be STYOOart. Originally,
Stuart was the name of a Scottish and English dynasty -
James Stuart,
James VI of Scotland and I of England, succeeded his aunt,
Elizabeth Tudor, who had no children. And that in turn was originally an occupation name, a stewart or steward being a senior servant/official in a royal or otherwise prosperous household.
My nephew is
Stuart, and spells his nn Stewie. Since the word 'steward' is still in use, it's my impression that
Stuart is the more usual as a fn.
I quite like the name (and the nephew!) but not enough to use.