Will, someone commented on the fact that you'd asked about music but got replies about song; this is probably because your original post mentioned the Copper family, who are purely vocal, so I'm referring to both music and song in this reply.

I think you need to get away from identifying music you listen to as sounding "celtic" or "non-celtic" or "celticish". There may be a sound you're looking out for but it's really not got much to do with the Celts, who ceased to be an identifiable race and culture many centuries ago. "Celtic music" is a marketing term, applied mostly to the modern style of Irish folk music (featuring vocals accompanied by bands, octave mandolas, etc).

Here are a few (very generalised) observations for you to think about:

Vocal harmonies are much more a feature of traditional English song than of Irish or Scots song. Traditional Irish song relies more on ornamentation. Irish music has its own distinctive sound; Scottish music has (to my ear) as much in common with English as with Irish. Highland Scots are (if you want to be racist about it) the celtic part of Scotland, and the music there is different from the mainstream, lowland Scottish music. English music, too, varies very much across the different parts of the country. Anglo-Saxon isn't necessarily the dominant feature - the Normans and the Vikings/Scandinavians have had a fair bit of input into English culture in some areas. And of course there is lots of overlap between English/Irish/Scots/Welsh music and song.

Just forget the misleading and unhelpful "celtic" label and try listening to a wide range of music from these islands, both the older traditional stuff and the more modern "revival" music and song. Although we're a very mixed race now, you'll soon begin to identify distinct regional or national differences in the sounds you hear, and I'm sure you'll find much to enjoy.