I love the song, and I still sing it. Children love to hear me sing it, especially because I am over weight.
I think it may be a parody on the "Alice Where Art Thou Going" archived at Indiana University as mentioned above. I sing:
Alice, Where art thou going?
Upstairs to take a bath.
Her body was like a toothpick.
Her head was like a tack.
Oh, Alice stepped in the bathtub,
Pulled the plug right out!
Oh! my goodness, Oh! my soul!
There goes Alice Down the hole!
Alice, where art thou going? (The End)
I'll try to write the tune for you. I think I sing it in the key of b-flat.