Norman French was never the language of the ordinary people; just that of the new aristocratic elite. The Normans were small in number. Essentially, they established themselves as a separate caste, which soon had the thrones of not only England, but also Scotland (Robert Bruce, for example, was Norman-French, not Scottish, by descent) and Wales.

It would seem that much the same had happened with the earlier Germanic immigration into Britain, though in that case the existing population had embraced the "new" language (though retaining a good bit of "Celtic" vocabulary -particularly placenames- and some structural elements). The "displacement" theory is some centuries old and now largely discredited, both on linguistic and (more recently) genetic grounds.

The main function of Norman French was in its mutating influence on "Anglo-Saxon", vastly widening its vocabulary and accelerating its development from an inflected to a non-inflected language. From Chaucer's time English in its "new" form replaced French as the Court language in both England and Scotland.

The term "Celtic", as applied to nationality as opposed to language or archaelogy, is relatively recent, gaining currency during the 18th and 19th centuries largely as a by-product of Romantic Nationalism. It is essentially meaningless as applied to music (and many other things): as has been suggested, a song or a person doesn't become "celtic" because they are found a few yards on the Scottish side of the (flexible) border, any more than they might cease to be if found a few yards on the English side. Lines on a map are not hermetic seals.

The whole idea of "separateness" is ridiculous. We live, and have lived for a thousand years and more, in a constantly evolving cultural continuum where national boundaries (including those with Ireland, I might add) have been geographical and political, but decreasingly (except where different languages have been involved, and much of the time not even then) cultural.

I'd disagree with "Nerd" on some of his comments (particularly where "irony" is concerned; there is a lot more to it than that), but that would be for another time.