My dad used to sing it to us (I'm 48 and my brother is 46) as:

Oh Mr Moon, Moon, lovely Mr Moon
won't you please shine down on me
Oh Mr Moon, Moon, lovely Mr Moon,
hiding behind that tree
Well when your life's in danger and you're on the run
There's a man behind you with a great big gun
Oh Mr Moon, Moon, lovely Mr Moon
Won't you please shine down on, never cease to shine on,
Moon shine down on mee--e--ee!


Don't know where he heard it--he died 5 years ago, but both my brother and I recall the lyrics the same.