Of course many xmas songs in english are translated into french. For example :
White Christmas = Noël blanc
Silent Night = Douce nuit
Winter wonderland = Au royaume du bonhomme hiver
Jingle bells = Vive le vent
O come all ye faithful = Peuple fidèle
The little drummer boy= L'enfant au tambour......

But one of the most famous xmas song here is " Petit Papa Noël " (= Little Father Christmas). All the children learn it when they are very young. Let me try to translate part of the words for you :

C'est la belle nuit de Noël (It's the wonderful christmas night)
La neige étend son manteau blanc (snow is spreading its white coat)
Et les yeux levés vers le ciel (and the eyes looking up to the sky)
A genoux les petits enfants (little children are kneeling down)
Avant de fermer les paupières (before closing their eyes)
font une dernière prière (are making a last prayer)

chorus : Petit Papa Noël (Little Father Christmas)
Quand tu descendras du ciel (when you come down from heaven)
avec tes jouets par milliers (with your thousands of toys)
n'oublie pas mon petit soulier (don't forget my little shoe)