I'm surmising that it's an old "music hall song" as opposed to "traditional" in the true sense of the word.

So, Sandman might be on the right track even if it wasn't Harry Lauder. I've no idea one way or the other.

Incidentally, there's some familiarity with this well known Irish ditty..

https://youtu.be/SViPByHqvk4

I'm not sure if the tune is "related" or not. of course. However, whoever wrote either tune first might have influened whoever wrote the second. ;-))