That was written for the movie. I think it was called "You're so beautiful" (not to be confused with Preston's song) and was sung by Greenwood himself. Beautiful song. I really love it
I suppose it's possible that the mystery song was written for the movie as well (by Jay Levy?). It would have been cheaper to have done it that way.
I think I came to a conclusion: You guys who said it was an original song were right! Listen to the music played in during the end credits and compare:
https://www.youtube.com/watch?v=MOI8UbRyxG4The outro was cut when the movie was shown on german TV in the 90s which is where I got my copy from.
Still, that doesn't explain, why I'm sure I heard it on the radio once. Maybe I had too much Tequilla....
That Jay Levy is a cool dude. I do like his original compositions for that movie very much!