My father, who died in 1963 used to sing only the bread & gravy verse, but changed the end to "on Friday we went to the mommy, can we have something else instead, so..."
Otherwise the sAme. He always played with lyrics anyway. He had been an usher in burlesque in Cleveland in 1928, so we all thought, of course - until I heard two women sing it at a big band music show here in Poughkeepsie, NY about ten years ago. Thought I'd fall off the chair and they loved it when I sang along. Still don't know where it comes from but loved reading the input from everyone. Of course, I sang it to MY kids, and will sing it to my grandkids.