Here's the scene as I see it: A man and woman have just gotten married and they're going to bed. As they start to get into bed, they probably each first sit on the edge of the bed on their own side, with their backs to each other. Then they get into bed and TURN to face the other person.
So , in the song lines, " turn round" means turn your body around to face someone( where previously your back was to the person.)