I suggested on the other thread that the song probably originated in the 1930s. Perhaps I was wrong:

"Bro. Buck Murphy brought tears to the eyes of some of the boys by his creditable rendition of that sentimental song, 'Only a sliver from my sweet-heart's wooden leg.' Bro. Tommy Ward rendered a very catchy song entitled, 'We had to move Paddy's grave to dig a sewer.'"

(From "The Journal of the International Brotherhood of Boiler Makers and Iron Ship Builders of America," Aug. 1, *1903*, p. 407.)