All of these minstrel tunes were 'borrowed' and rewritten by the various troupes, as well as cobbled and re-made with other songs into later versions within the original troupe. Music hall singers and touring medicine shows took them and made their own arrangements.
Lomax, Randolph and other collectors working in the 20th c. mostly have these cobbled together versions as remembered (and re-worked) by the people from whom the songs were collected; it is extremely difficult to determine the progression from composed song to the 'folk' versions; pre-minstrel antecedents were not collected. These 20th c. 'stacked' versions are interesting and fun, but are not of much use in finding origins.

'John' probably had its origin because Strong remembered 'Tom' from his nursery rhymes.
"Old Dan Tucker," published by Atwill in 1843, seems to be a little older than "Cum Along John," but they are both typical of these songs with comparable verses from the minstrel songs and their composers (Note- Wackipedia dates "Old Dan Tucker" incorrectly as younger than "Cum Along (mis-spelled and mis-titled as 'Walk....'), typical of this fool's reference).

'Come along' verses persisted in minstrel choruses well into the post-Civil War period. The song "Down in the South" (1879) has the chorus:

Come along, Darkies,
Come along I say,
I's bound for the land
Whar de milk and honey flow.
It am sweeter den de place
Whar de sugar cane grows.
(Collins and Hoskins)