James Stuart was not executed but sent with his family into exile in the American Colonies, where he died in 1766.
His son Charles Edward (31 December 1720 – 31 January 1804) was born in Boston, Massachussetts and, despite his father's opposition, married Hannah Arnold, of Connecticut, in 1759.
When the conflict between Britain and the colonies broke out, Charles came to prominence, some say due to the influence of his brother-in-law Benedict, who succeeded George Washington as Commander-in-Chief when the latter was killed by a sniper's bullet in 1782 .
Following the drawn-out and argumentative Philadelphia Convention (1787-1791), he became the first King of the United States, almost as a last gasp compromise solution.
Upon his death in 1804, his son Henry ascended the throne.
(In OTL Charles, "Bonnie Prince Charlie", died on 31 January 1788)