I believe the tune used was "Scotland the Brave" but it's been a long time since I saw the movie. There was a popular release about that time called "My Bonnie Lassie" to the same tune - and there was also about then a mildly popular version of "Green Hills of Tyrol" by Andy Stewart. Could that be it?
"there was a soldier, a Scottish soldier,
He travelled far away and soldiered far away.
There was none bolder, with good broad shoulder
He fought in many a fray and fought and won
But these green hills are not highland hills
Oh the island hills are not my land's hills..."