Luckily in 2007, Leslie Mahler wrote an article on his English origins. I very much enjoy reading Mr. Mahler's articles (and he is prolific). In this case, the clue is sitting right there and has been in print since 1905. Richard Olmstead, of Norwalk, Conn., leaves legacies in his will to his cousin Bazies, his two daughters Elizabeth Peck and Lydia Baker. Further in 1912, a Slaney will was found that connected both families. Using those two things, Mr. Mahler found John Baisey's father's will--a typical English will, not in the Prerogative Court of Canterbury, and one in which he neglects to note that his son John was living in New England. So, in this case, it is not the father's will that proves the origins, but two cousins' wills (and a rare surname in that part of England).
John's wife Elizabeth's surname is still not known. They had three daughters: Mary who married Samuel2 (Benjamin1) Burr, Elizabeth who married Paul2-1 Peck, and Lydia who married John1 Baker. The full article is: “The English Origins of John Baisey of Hartford” by Leslie Mahler The American Genealogist 82 (2007):32-38.
It should also be noted that there is no genealogical cross reference index for John Baisey being mentioned in the Olmstead Genealogy. However, the first part of the Olmstead genealogy appears in the New England Historical & Genealogical Register which does have a print index and is available online. Otherwise, how would one connect those two families?