Hi Kate,
Do you happen to have the records for the Bowl Alley Lane Presbyterian or Unitarian Church? A number of Fairbairn/Fairburns were baptised there between 1769 and 1780 with father, John. From the IGI - Betty 1769, Sally 1772, Nancy 1774, Nelly 1777, Robert 1780 and John 1783. Is there a mention of a mother's name?
Thank you.