"Colly" is derived from "coal", a word at least as old as Shakespeare. "Calling" birds is a later alteration. Use of coal for domestic heating in England became more widespread in the 1500s when the wood supply came under pressure, but the Romans were mining it there already in the 100s.
"The Twelve Days of Christmas" dates to at least c. 1780, as shown here from "Mirth Without Mischief", perhaps but vaguely recollecting the traditional drunken revelry of Christmastide which English Puritanism had long attempted to mute. This work was for children, who were in any event prone to mischief after school.
