This lovely tune, remarkable for its haunting Mixolydian mode form, is unlikely to be 12th century. The words are maybe older than the tune, which likely came from the wave of Irish Music during the late 18th and through the entire 19th century, American Civil War period. Such rhyming words and tune structure had not been invented sufficiently in the 12th century, and it looks more like the 16th if it can be proven to be earlier than the 18-19th.
The Mixolydian mode can be played with only two chords one step apart on lute or guitar, which makes it an easy and wonderful performance piece for good singers but beginning players. Celtic Woman does it in D and C chords, though there are variations out there.
For the most likely oldest Christmas hymn, I suggest it may be "O Gladsome Light" with words from the 3rd century, tune possibly the 8th as a Nunc Dimittis. Source "Handbook to the Lutheran Hymnal," by Pollock. Also the still popular "O Come Emmanuel" may have been written about that time. The main problem is that these ancient Christian tunes, unlike the Celtic tunes, usually had a long oral tradition in the church before musical notation became understandable around the 14th century.