It is now clear Nathaniel Cooke (1621?-1688, Windsor, Conn.) was not related to Aaron Cooke (genetic testing shows them to be in entirely different haplogroups).
It has been suggested that he was the Nathaniel Cooke baptized 20 March 1620/21, in Crewkerne, Somersetshire (the same town where his wife was baptized). He was son of John Cooke (alias Hooper) and Rachel Daubeny (married 1602), and they likely knew the Vore family in Crewkerne. I have not yet been able to determine if they ever came to New England. If their son Nathaniel was the immigrant, perhaps he instead came alone to Windsor, Connecticut, shortly before his marriage to Lydia Vore in 1649, possibly because he was related to Rev. John Warham (whose mother was also a member of the Cooke "alias Hooper" family of Crewkerne).
Does anybody have any opinions whether Nathaniel (b. 1621), son of John Cooke and Rachel Daubeny, was Nathaniel Cooke of Windsor, Connecticut (who married Lydia Vore)?
"England Marriages, 1538–1973 ," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:NKCR-BY9 : 10 December 2014), John Cooke and Rachell Daubeny, 21 Oct 1602; citing Crewkerne,Somerset,England, reference , index based upon data collected by the Genealogical Society of Utah, Salt Lake City; FHL microfilm 0481077 IT 2.