I can't actually confirm the dates for those children listed as you have them. But I have found marriage records for several of the children of John Gately and Jemima Mercer in Caldwell County, Ky. Jemima gave her consent as mother for both Sally's and Jemima's (Jr.) marriages. These marriage dates indicate the birthdates of the parents to be approximately correct.
Sally Gately md. Aaron Smith on 7/7/1812 Caldwell Co. with Wm. Thornton as security and Jemima's consent Jemima Getely (Jr.) md. Samuel Bradshaw on 7/20/1819 Caldwell Co. with John Gately as security and jemima's consent
I think this 3rd marriage is yet another of Jemima's daughters. However in this case a Wm. Atchison made oath that Nancy was 21.
Nancy Gately md. THomas james, Jr. on 9/15/1813 in Caldwell Co.
I think Jemima located to caldwell Co after her husband's death because her brother JOhn Mercer was located there. Data seems to suggest he was the same John Mercer (Messer) who had previously been in Laurens Co., SC.
I think it very odd that John Gately and Jemima "took in" these 3 Bush orphans only 1 1/2 years after their own marriage. And to what advantage would the orphans be to Gately at their young age. I think that it very possible the orphans were related to either Gately or Mercer.