I'm so pleased to have found that others know this song.
My granddad used to sing it to us when we were children.

In his version, the lyrics are:
Goodnight, Mr. Moon, we will have to go and leave you,
Goodnight, Mr. Moon, we will have to leave you soon.
And when we're asleep, through the window pane you'll peep,
We'll wake up and call out, goodnight Mr. Moon.

He only ever sang it to us a capella, but I could imagine what the accompaniment sounded like (to me). I worked out some guitar chords for it, perhaps someone else could comment on the melody/harmony.

D A | D A | F#m D | G Baug-B7
Em B | E C#dim | A Aaug | D A
D F#m | Am D | G Gm7 | Bb Bbdim
D D/C# | D/C G/B | Em/G A7/C# | D

I'll have to find out how to write the melody without manuscript paper!