I had never noticed this before analyzing the carol for this post, but the ivy is mentioned in the first line and then it just disappears. Here are the first two lines:
The holly and the ivy, when they are both full grown,
Of all the trees that are in the wood, the holly bears the crown.
Wouldn’t you expect that there would be a third line explaining the role of the ivy, something that starts out with “Of all the vines that are in the wood, the ivy bears . . . ” But there isn’t. Here’s somewhat of a explanation from an academic website:
The lyrics are somewhat puzzling. The first line is “The Holly and the Ivy,“ yet ivy is mentioned nowhere else in the carol except in the last verse, which is a repeat of the first verse. Holly is given the starring role in the song and ivy is ignored, so it seems strange that ivy is even mentioned.
The explanation that is often given is that the first line in the carol is a remnant of the old custom of linking holly and ivy together. In the rest of the carol ivy isn’t needed. The “holly” in the carol refers to Christ and the theme of the carol is his life. (“English Ivy Symbolism, Traditions, and Mythology“)