The words and the music of "O come, O come, Emmanuel" developed separately. The Latin text is first documented in Germany in 1710, whereas the tune most familiar in the English-speaking world has its origins in 15th-century France.
Because "O Come, O Come Emmanuel" is a metrical hymn in the common 88.88.88 meter scheme (in some hymnals given as "8.8.8.8 and refrain"), it is possible to pair the words of the hymn with any number of tunes. The meter is shared between the original Latin text and the English translation.
However, at least in the English-speaking world, "O Come, O Come Emmanuel" is associated with one tune more than any other, to the extent that the tune itself is often called Veni Emmanuel.
Day 1 - "O Come, O Come, Emmanuel" - Musical Advent Calendar