You’re correct that the 1897 Saltonstall genealogy by Leverett Saltonstall misstates the relationship between Sir Richard Saltonstall, Lord Mayor of London, and Sir Richard Saltonstall, the Massachusetts colonist. They were not uncle and nephew as is commonly stated. The 1897 work was superseded by more detailed research done in the 1920s by J. Gardner Bartlett, who was able to make better sense of the many Richards and Gilberts in the family. His work, “Historical Genealogy of the Saltonstall family in England and America”, was never published (he died in 1927), but copies of his manuscript are held by a couple of libraries in New England (including the NEHGS) – and the FHL has a copy on microfilm.
The common ancestor of the American Saltonstalls and the later English Saltonstalls was Richard Saltonstall (d. 1524) of of High Saltonstall and Shelf in the parish of Halifax, Yorkshire. His third son Gilbert (b. ca. 1493, d. 1545), of Hipperholme, was the father of Sir Richard the Lord Mayor. His second son John was the ancestor of Sir Richard the colonist, as shown here:
John Saltonstall (b. ca. 1492, d. 1559) of Hipperholme
Gilbert Saltonstall (1525-1598) of Rookes Hall
Samuel Saltonstall (1562-1613) of Roger Thorpe
Sir Richard Saltonstall (1586-1661), the colonist
The confusion about the relationship of the two Sir Richards arose because the 1897 genealogy confused Gilbert (d. 1545) with Gilbert (d. 1598) and assigned Sir Richard the Lord Mayor as the son of the latter Gilbert rather than the former – thus making him the uncle of Sir Richard the colonist. This genealogy is also laid out (although not in full detail) in volume 1 of “The Saltonstall Papers, 1607-1815” (Robert E. Moody, ed.), published by the Massachusetts Historical Society in 1972, which cites the Bartlett work and notes the correct relationship between the two Sir Richards.
Curiously the ODNB biography of Sir Richard Saltonstall the colonist, while citing both the Moody and Bartlett works, continues the error of making the colonist the nephew of the Lord Mayor. Presumably the text was carried forward from the old DNB without adequate checking by the author of the ODNB article. (The ODNB article on the Lord Mayor, by a different author, does not mention the colonist.)