Well we've come a long and interesting way.
Thanking everyone here, especially Azizi, for heading me in the right direction. That is on to several of my old books, the Bodleian Ballad Collection, and the Lester Levy Collection.
Here are my conclusions about my original question.

"Jim Along Josey", Published in 1840, and sung by minstrel troupes of the day, has the verse:

My sister Rose de oder night did dream,
Dat she was floating up and down the stream,
And when she woke she 'gan to cry,
And de white cat picked out de blackcat's eye.

This song has a chorus that is similar in phrasing to the tune -- "The Black Cat Piddled in the White Cat's Eye".

My considered opinion is that "Jim Along Josey" was probably used by fife and drum bands in America. Before, during, and after, the Civil War.
Also that it would have been known in Australia. Brought by the minstrels.
The colour of the cats could have been changed to suit circumstances.

There are several variants of the phrase -- "And de white cat picked out de blackcat's eye":
"Black cat piddled on the white cat's tail".
"Black cat kicked out the white cat's eye".

The tune "Black Man Piddled in the White Man's Shoe" was collected from New South Wales. Alan Musgrove diddled it to me over the phone. It sounds nothing like the others we're discussing. Nor does it sound anything like "Old Dan Tucker" although it is also called by this name.

The great Sam Cowell may have written "Jim Along Josey" but the hard evidence is lacking.
Cheers, Joy