I would love to get this song from you including the variation. Thank you very much. Please would you also include how the song was known to be sung? You are welcome to email me directly at firstname.lastname@example.org.
I too thought that the navvys may not have composed songs for the same reasons as you cite. I would imagine the emigrants that were handed $5.00 and conscripted to fight for both sides in the Civil War were not of any greater literacy and on balance had no more time than the navvys and their songs from the era survived.
Who knows really but I have faith that one or two will turn up if there is enough digging (ironically enough).