Now I've got it stuck in my mind that it was about the same time as the Sassenach invasion of the South.
You could well be right, faldage. As I posted, I had a sinking feeling that it was, in fact, post-Roman - we need a Dark-Ages historian to help us out here. My C19 expertise is of little help and I freely acknowledge that i am well out of my period, here!

As to the Gaeltag "conversion" of the southern parts of Scotland, I'm not sure. Certainly by the Early English period (late-C15 - mid-C17) the Scottish version of English was the norm in the lowland areas and Gaelic was in the wild and uncouth Highlands.

Control of the northern parts of England and the southern parts of Scotland passed from one tribe to another throughout the dark-ages and the mediaeval period, anyway. Lancaster Castle, in the city where I now live and which was a building in which I taught for some years, was started by a Scottish King (David II, if I remember correctly) in ca. 1150. and the voillage in which I lived, some six miles south of Lancaster was right on the Anglo-Scottish border for about fifty years.

to Dub-dub
The invasion to which I refer is when the indigenous unhabitants of Ireland (Hibernia), the Scots, (believe it or not) invaded the northern-most areas of Britain (Caledonia), driving out the indigenous inhabitants (the Picts) and calling the area taken over Scotland.