Found 1880s in Harvard songbook, according to the Traditional Ballad Index, and as noted by Masato.
The problem with making any attribution of these simple rhymes with simple meters is that they are common to both black and white cultures, and floated among dance and party songs of both groups. With few exceptions the rhymes were collected in the 20th century, except for those in the minstrel songbooks (and in this case, school songbooks).
Even in the 20th c., collectors like Randolph, Cox, Belden and others mostly ignored them or did not collect them, except for a few like Old Dan Tucker and Buffalo Gals which, as far as we know, mostly derive from minstrel roots.

That there must have been many among whites is evident if one goes through websites such as the one hosted by the Bluegrass Messengers for fiddle tunes and lyrics: Fiddle Tunes
Here again, few collections were made of rural dance music until well into the 20th c.

These lines are found in "Cum Along John" and other minstrel songs of the 1850s.
Milk in de dary nine days old
Rat an' skipper gettin' mity bole,
Long tail rat in a pail ob souse, etc.

Dey go to bed wid all dar clothes on,
Dere legs hang out for chicken to roost on.

Ole cock crows, de young wuns larn

Opossum up a gum tree
His tail has body follow

Way down souf on de beever kreek
etc.