I'm not sure to be honest. The spelling Steuart was used in 16th century England.Stuart' was a French spelling, right?
The vowel /u:/ 'oo' was becoming /ju/ 'you', so the spelling Stuart could have just reflected that.
I think the Stuart families in France claim descent from various branches of that dynasty
when it was kicked out from the UK, it wasn't only royal family that upped sticks and departed.