I - vi - ii - V- iii - IV - IV (or ii6) - V - I.
The pattern breaks at the last dominant chord but none the less, I found the pattern that was there interesting. Perhaps the pattern contributes to why it is such a popular carol. By having a pattern in the underlying harmony, it is easy to tune and sing a cappella because your ear can anticipate where its going. Also, the melody has a similar gradual outward expansion in pitches, starting at the tonic. By warming up the singers' ranges throughout the piece, no one has to be anxious about the larger interval leaps.
Also, I should mention that the vowels in phrases such as, "hap-py new year" lend themselves to the pitches that are assigned. The [i] vowel slightly raises the pitch, while vowels with a flatter tongue such as [æ] and [u] lend themselves to lowering pitch.
If anybody has their own finding or finds/knows something contrary to what I found, feel free to share!
and Happy Holidays if you celebrate~
Prayers of health and prosperity to all.