Hello Jacqui
I would expect your couple simply walked over to Scotland to marry either with their Presbyterian minister doing the ceremony or by some less formal ceremony. Border or irregular marriages were very common in northern Northumberland amongst working people. There May be no surviving record especially at that date.
All marriages in England between 1754 and the start of civil registration in 1837 had to be in the Church of England for several reasons but including problems with record keeping. There were exceptions for Quakers and Jews who kept their own excellent records. Scotland had different rules and many ordinary Northumbrians, even some who were C of E, simply walked over the Border for a cheap and discreet marriage which was legally binding though irregular.
See below for some info or google for Border or irregular marriages. The two booklets published by the NDFH also have good introductions giving the background- your marriage falls outside the periods covered by the booklets.
https://www.gla.ac.uk/schools/socialpolitical/research/economicsocialhistory/historymedicine/scottishwayofbirthanddeath/marriage/P.s. I have assumed you have checked Scotlandspeople, familysearch.org and FreeReg and FindMyPast for a marriage for your couple?
I see an Adam Winter was bap in 1751 in Tweedmouth and a 48 year old Adam Winter buried in Tweedmouth in 1799. FreeReg says he was a shoemaker. (Thanks ticketyboo as I see you were the transcriber). If this is your Adam, have you checked Tweedmouth for any additional baptisms which may give you his wife’s name if you are very very lucky?