In the NDFHS journal for spring 2011, there was a very good article about presbyterianism in the Northumberland highland areas, such as the North Tyne and Coquetdale. I am pretty sure that that was where I read about the Presbyterian Church in the area being known as the Scotch church.Certainly a lot of ministers in that area were Scots and there is a tradition that it became established when Covenanting ministers came south over the border to escape persecution. There was a Scotch Church in Berwick, at least in the 1st half of the 19th century and also 2 in Alnwick. Whether these could formally be classified as Church of Scotland is another matter. St James in Northumberland Road, Newcastle, (think that's right) was Church of Scotland at one time, but broke off formal links in the 1830s. There were about a dozen Presbyterian churches in Newcastle in the late 19th century, and some of these may have had Church of Scotland links.
I have a vague memory of an elderly friend of my parents, born at the end of the 19th century, talking about her Scottish parents taking her to a Church of Scotland church in Newcastle on Sundays, and my memory is that it was in the Northumberland Road area. Looking on old lists of churches for Newcastle, I see there was a Trinity Church in Northumberland Road, but I don't know anything about that. Or perhaps it was St James's and a lot of Scots happened to go there.