Hi Phil, showing my ignorance now, I did actually think Allingham was later as I seem to remember he did a derivative book of ballads, but may be wrong. I realise now you were trying to give his dates which were 1824-1889, so the date of the poem is c1853, so he could easily have been influenced by the chanty and it doesn't tell us much about it except what we already knew, the chanty was popular by the 1850s.
Still interesting though, and thanks for posting it.