Thank you for all this helpful information on the song. The reason I arrived at your site is that in the collection of the Frances Lehman Loeb Art Center at Vassar College, Poughkeepsie, New York (where I work) is a painting by the American social realist artist Ben Shahn, entitled "Pretty Girl Milking a Cow". It depicts a workman sitting on a hillside playing his harmonica. No girl nor cow in sight. Thus, we assume, the title of the painting refers to the tune he's playing. The painting was done in 1940 so I wonder if there is a possible connection with the popularization of the tune thanks to the musical film "Little Nelly Kelly"?

Wish I could attach an image of our painting for you to see.

-James