So just to be technical, there has been no English or British monarch named James III. James Francis Edward Stuart, the exiled son of the last Stuart king of England and Scotland, styled himself as such, but he was never crowned. Hence his historical nickname, the Old Pretender.

I have seen no evidence that would indicate close African ancestry in James Stuart. This is a portrait of James Stuart as a young man. This is his father, James II, before being removed from the throne. His mother, Maria di Modena, was the daughter of the Italian Duke of Modena. If these portraits are even mostly accurate, I don't see any reason to assume James Stuart was anything other than a white man.