"it is an example of a passamezzo antico, an Italian minor key ground bass or chord progression, upon which a composer would fashion a tune."

I could say that of any tune whatever that started with a minor chord, any tune from a Gregorian chant to a TV commercial. There is no reason to associate Greensleeves with anybody Italian.

If Henry VIII or any other noble person had composed Greensleeves, his secretaries, courtiers and hangers-on would have left us reams of fulsome, doting praise. The tune was probably written by a commoner and musician, a person of little importance at that time.

I believe Greensleeves was written by somebody Celtic because it probably wasn't about sleeves but about 'sliabh' a mountain. There are lots of songs about mountains, but whoever heard of anybody sighing over his lover's sleeves?   Other musicians quickly learned this fine tune and spread it the length of Britain. In time, many sets of words were written for it, but the idea that it's really 'Greensleeves' has persisted to so long and so strongly that I believe it is the original title.   

Don't ask me to define Celtic. Asking somebody to define Celtic is like asking somebody to define purple or chocolatety. We all know what it means - enough for this discussion, anyway.

By the way, O'Neill's Music of Ireland has a 'Green Sleeves and Yellow Lace' which is a the same tune in jig format. It sounds Irish. But who knows whether it went from Ireland to England or the other way, after all this time?