I guess the most obvious besides the ones in the OP is “Ave Maria”. This was not originally a Christmas carol; its original title is “Ellens dritter Gesang” (“Ellen’s third song”), and it was written by Franz Schubert as part of a suite of songs based on Walter Scott’s poem The Lady of the Lake. It was later used as a setting for the Latin version of the prayer “Ave Maria”. Here is Joan Baez singing it in German; I’m not sure if these are Schubert’s original lyrics or not.
The carol commonly known in English as “The Carol of the Bells” was originally written by a Ukrainian composer; it is based on a pre-Christian folk chant, “Shchedryk”, which celebrates the coming of the New Year (which was traditionally celebrated in Spring). With the coming of Christianity, the New Year was moved to January, and the holiday the chant was associated with was changed to Epiphany. The English lyrics (written in the 1930s) are completely unrelated to the original Ukrainian ones. Here is the Ukrainian version.
Another non-English carol I like is “Pat-A-Pan”, which is Burgundian in origin. Unfortunately the only clip of it I could find with the lyrics in French is by David Archuleta. If you must, here it is.
Another French carol is “Cantique de Noel”, known in English as “O Holy Night”. Here it is sung by Joan Baez.