As far as I understand, the question is : 'Amazing Grace' is or not a Celtic song. But all the answers are from the question : Where this song come from ?
As Breton coming from Bretons for more than 500 years, I consider myself as an actual and contemporary Celt, descendant of the first settlers in GB returned on the continent. Anyway, though this song is overfamous in all Great-Britain and the USA, there are Celts who don't know it : in Brittany, I didn't heard about it before my 15's, when I involved in traditional music, and it cames to me through the Scottish bag-pipes. Since the renewal of the Celtic identity in Brittany, Scottish songs are more and more famous. But this began only 30 years ago.

I just would like to understand what is a "Celtic" song ? The songs and the dances of my fathers in Brittany have no way in common with those in Ireland, and both our Breton, Irish & Scottish music is far from the music popular at the age of the Celtic expansion, 600 years BC.
In fact, what makes a song "Celtic" ? Its popularity in Celtic countries ? The fact that the oldest known version comes from a Celtic isle or a reknown Scottish musician ? You know that the music has no borders.

OK, those questions are for historians, musicologues, etc. Give me a help if you know some answer. But as ancient music player, I'll be unable to split a music between Celtic and non-Celtic. So : Is 'Amazing grace' Celtic ? I don't know, but I like singing it !