Randolph and Legman, in "Roll Me in Your Arms," have a long discussion of this song, which they put under the title "Tickle My Toe."
Jingalo seemingly is from Zingaro, a gipsy. They consider the song to be of British origin, 18th century at the latest.
"The Corn Grinds Well" (The Miller, Mathew the Miller) was printed in part in William Chappell, "Popular Music of the Olden Time," 1859, vol. 2; Reeves printed further words in "Everlasting Circle," collected in 1906. Cecil Sharp's "Gently Johnny My Jingalo" was collected in 1907, and published in much expurgated form.

As Malcolm Douglas said earlier, there are many versions; the first one noted for North America in the 1890s. Ed Cray, in "The Erotic Muse, American Bawdy Songs," discusses the song under the version title, "Billy Go Leary."
"He Put His Hand Upon Her Toe, Yo Ho, Yo Ho," was a version sung by marines.