The melody for the quintessential Hannukah song - Maoz Tsur (Rock of Ages)is taken from a German folk song first utilized by Luther.
Woody Gutheri also wrote some Hannukah songs:
And now Senator Oren Hatch hit the jackpot.
I cannot imagine any downside to this type of interfaith collaboration. Still, Jews who write Christmas songs emphasize its non religious aspects. To honor my Christian friends who like to emphasize the religious one, here is this year's Cinderella - the number on record seller in both Britain and the US in a song not written by Jews.