I've wondered for years if anyone else had ever heard this song and its origin.

In the later part of the war my friend's father used to sing it to us.
Here is his version of the chorus. (I've always thought that it was the whole song.)

Our house is the first house,
And the next house is Mr Waterhouse's house,
Waterhouse's house is next to our house,
And the next house is Waterhouse's daughter's house
Next door to Waterhouse's daughter's house is
The public house
The picture house
The slaughter house
And in case you want to know,(pause)
The last house in the row, (pause)
Is everybody's house (pause)
And that's The Workhouse

I really feel that this sounds more authentic, given the play on the word "house", making it more like an Olde Thyme Music Hall Song and a bit of a tongue twister.