I would challenge you to find any animals in the actual nativity story of the Gospels with the exception of the “flocks” over which the shepherds are watching in the fields. We usually assume that these were flocks of sheep, but there could have been goats too. Other than that, I regret to inform you that those cleaned-up, prettified little animals that show up in manger scenes almost certainly weren’t there—with the exception, perhaps, of some ewes and newborn lambs. So when the little drummer boy says that “the ox and lamb kept time” while he played his drum for the Christ child, we can be quite sure that this is a later interpolation into the story, at least as far as the oxen are concerned.
Regardless of any scriptural basis for animals in attendance, there are surprisingly many carols that include animals, and not just as window dressing. Below is a selection of carols that include birds. (My own choir, the Cherry Creek Chorale, is singing a couple of them for our December 2024 concert. Follow the link if you live in the Denver area to get your tickets!)