I think the question is essentially a bit meaningless. For example, your staement that Celtic music is instantly recognisable as such only refers to a very recently created modern sound(eg Pogues, Enya, Chieftains) which is basically an Irish/Scottish modern sound, heavily influenced by the bRitish folk revival of the 60's (which, in your terms, would be Celtic and Anglo-Saxon since).
   Now, what I am saying is, I dont think you would find actual Celtic traditional music so instantly recognisable. If you find the oldest possible traditional recordings of music from say Ireland, scotland, Wales, Cornwall, Britenny and Galicia, and have a listen. you would not find this common "Celtic spund" so easy to spot. You might find a blindfold test quite difficult to pass, if a bit of music was thrown in to fool you from places not cenventionally thought of as "Celtic".
   And anyway, it is just plain bad history to refer to the inhabitants of Ireeland and Scotland as "Celtic", and the inhabitants of England as "Anglo-Saxon".
    So personally, I wouldnt like to dig into to any argument on the"Celtic sound" in music, as in the last analysis I dont think there's really any such thing.
    But the differences, and similarities, of music in the various regions of the British Isles(and northwestern Europe in general) is a topic I'd happily get into.
    As regrads the sounds of English music, there is a considerable variety from region to region(or was). As in Ireland, some of these regional differences are being smeared out fast by the structure of the folk scene, population movement etc.
To start with what I know about: my own band's new CD "A Trip to the Lakes" by the Boat Band, is devoted to NW English musicwhich borders on Scotland and Ireland(via the Iriah Sea). A quick look on google will find tracks you can download (some for free)to get a flavour. You can also find, on the internet, my aricle about William irwin, the Cumbrian fiddler, which I hope you find interesting as background.
   And William Irwin provides a fine example of why I find the initial Celtic/Anglo-Saxon structure of your approach so unproductive and confusing. William Irwin was an English fiddler, so in the "Anglo-Saxon" camp in your theoretical world. But Cumbria, the part of England where he lived, means "Land of the Welsh" or "Land of the Celts". And his surname, Irwin, might suggest a possible Scottish family connection. But the majority of placenames where he lived were Scandinavian (predominantly Norwegian). Reality is often much more complicated, and much more interesting, than simplified modern theories.
   Have a listen to a couple of tracks, and tell me. Do they sound "Celtic"? or "Anglo-Saxon"? Or something else? And if so, what?
Thanks for raising a very interesting topic.