Mary Queen of scotts was the first to use the Stuart spelling of the name. King James who authorized the English Bible bearing his title is sometimes listed as James Stuart but more often as Stewart. Actually it is the same name.
There is a Stuart of Bute family which settled along the Irish Sea in lowland Scottland and presumably adopted the Stuart spelling to simplify relations with the Irish who have no W. Ulster Scotts also adopted that spelling for the same reasons.
In America many Stuart's or Stewarts would select the spelling that fit in with their origins. Those transported around 1735 were more often Stuart but many Stewart families changed with time. And of course many Stuart families began using the Stewart spelling. Any genealogy research has to be done with the thought that spelling was often the result of education and as I learned early on teachers think they know everything.
Alan Stuart aka Allen Stewart, Allan Stuart, Allen Stuart or Allan Stewart. And in WVa. Al'n Sturrit