I have no doubt that more Irish tunes than "Callino" were known in England in the 16th century, and I've made a case for "Bragandary" and cited evidence that the McPeake's "Monaghan Fair" was that sung in England about 1597, as "Derry's Fair", (both on my website) but without evidence I take speculations to be just that much more hot air to be ignored.

The earliest copies of "Greensleeves" and "Callino" are in lute tablature, as are many other 16th century tunes in England. What tablature was used for violins in 16th century England and where are examples of such to be found?