Is it possible the song with "upstairs to take a bath" started out as a parody of the following song? I don't know the tune that has been discussed here, so I hope someone who does will listen to this:

There is a recording, taken from a 1907 Edison cylinder, of Billy Murray singing ALICE, WHERE ART THOU GOING? at the Cylinder Preservation and Digitization Project. (Or, if that doesn't work click here to play the mp3 file.)

From the sheet music at Indiana University:


ALICE, WHERE ART THOU GOING?
Words, Will A. Heelan. Music, Albert Gumble.
New York: Jerome H. Remick & Co., 1906.

1. Her Christian name is Alice and her dad's a president.
Her home is not a palace but an East-side tenement.
It's at our labor meetings that her dad's chief officer.
He's there with friendly greetings, so I guess it's me for her.
Each Sunday morning dressed up swell, my visits never cease.
I ding-a-ling her front-door bell, and speak my little piece:

CHORUS: Alice, where art thou going?
Where can we spend the day?
Alice, we'll make a dead swell showing
If it costs my whole week's pay.
My coin was made for blowing.
Girlie, it's up to you:
Sail or ride or roam the sands,
Or sit and listen to the bands,
Alice, where art thou going?

2. If she lived in the White House or a shanty out of date,
Her home would be a lighthouse that would always steer me straight.
That sounds just like a poet and I like it too, somehow,
But if she was to know it, I can hear her laughing now.
Well, anyway, she waits at that, as sure as Sunday shines,
And blushes when I tip my hat, and speak my little lines: CHORUS


[There is a still older song called ALICE, WHERE ART THOU? that this might be a parody of. More investigation will follow.]