Love the South Park classics at Christmas. I wont even try and post a link to Mr. Garrison's Christmas song. 

Seems to me that what you want is to involve the children and make the learning fun. I would go for just the 12 days of Christmas and make a big production of it.
Sheet music:
kididdles.com/pdf/twelvedaysofchristmassheetmusic.pdf
 
Write the elements of the 12 days (Partridge in a Pear Tree, Two turtle doves etc) on cards. Forget the guitar. Just sing and hold up the card so that the kids can join in. Start slowly and then speed up as you have more of the 12 days to sing. Then give the cards to twelve of the children and have them try and hold them up at the right time (perhaps teacher might tap each one gently on the shoulder to remind them!). Everyone dissolves into giggles as teacher dashes back and forth along the line of kids with cards and great fun is had by all - and they learnt some not very easy English words! Afterwards you could also get the children to draw on cards what each of the 12 days actually look like - give a prize to the best.

Around 1980 or thereabouts, my sister was teaching ESL (English as a Second Language) to recent Vietnamese immigrant kids. In those days I often played Santa, had my own red suit, boots, theatrical makeup and round wirerim glasses and the "ho ho ho" down pat. So she invited me to come to her classroom and bring gifts, to introduce them to an American tradition.
 
I walked in, and the response was absolute terror from the children. They backed away from me, seeking the hands of older siblings while never taking their saucer eyes off me. One brave girl got close enough to pass candy and toys to the others, but they didn't even look at the goodies until I'd left the room.
 
As I made my exit, I could hear "We Wish You a Merry Christmas" behind me. At least they could relate to the Happy New Year part, even if the Santa experience had been traumatic. Those kids would be in their 30's and 40's now. I wonder how they explain Saint Nick to their children.

Santa scared the bejebus out of me as a child.  The concept freaked me out.  Some old creep wanted me to sit on his lap and beg for toys, while letting me know he was an omniscient peeping tom who sneaks in my home wearing PJs.  I'm not even joking.  I still have a picture of them trying to sit me on his lap and me screaming. i was brough up in a culture that loved Santa and i feared him. i can only imagine how scary that would be for someone new to the concept.

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Here's one:
 
http://soundclick.com/share?songid=4811022
 

Happy Saturnalia to you.....
 
there's lots of songs that retain the pagan origins of the festive holiday.....
 
they have already been listed so I shant repeat them here.

Guitarhacker
Happy Saturnalia to you.....
 
there's lots of songs that retain the pagan origins of the festive holiday.....
 
they have already been listed so I shant repeat them here.

But paganism is a religion 

musicman100

Guitarhacker
Happy Saturnalia to you.....
 
there's lots of songs that retain the pagan origins of the festive holiday.....
 
they have already been listed so I shant repeat them here.

But paganism is a religion 

 
Ah yes... you have a point.... but there has to be some sort of reason for the season, and, for the OP's purposes, at least they're not singing about the christian things that are supposedly related to the holiday.
 
Deck the Halls, for example is not a Christian song..... it's a Holiday song..... filled with references to the pagan side of the holiday.... the revelry, the yule log, the greenery.... without that.... it's not any more special than Dec 24th or the 26th, well, maybe not those days, because as I understand it.... the yule event celebration pagan thing went on for the better part of a week or until they ran out of mead.... or Jan 28th.... unless one of those happen to be your birthday.

I think December 22nd is the best day.
('Cause it's my birthday!)