This verse probably referred to Ring-Necked Pheasants, or perhaps alludes to Golden Pheasants. Both of them are natives of Asia but have long had successful populations in Europe and the British Isles. The Romans probably introduced them to Europe during their Empire. Pheasant were accepted residents of Britain by the 10th century.
Pheasant has a long culinary history, probably since Neolithic times. It is a popular game bird, today perhaps the most hunted bird on the planet.
Ring-necked pheasants were introduced in the late 19th century first in Oregon, where they succeeded on the second attempt. After that, they were introduced in other states and are now the state bird of South Dakota. This photo comes from that state’s department of tourism.
Golden pheasants are successful feral residents in England, but they probably were not introduced there until later than the carol, perhaps as late as the mid-19th century. Their astonishingly beautiful plumage could certainly have inspired songs about golden birds!
The bird second from the top in this painting by J.C. Harrison is a golden pheasant. The other pheasants are a Reeves Pheasant, at the bottom, an Elliots Pheasant at the top, and an Amherst Pheasant in the center. The black bird at lower right is a Mikado Pheasant. The bird at top right is not identified at Gamebird, but appears to be a Copper Pheasant. Prints can be purchased from Gamebird magazine at http://www.gamebird.com/pheasantprints.html.