Fwiw, I've not heard or read of this children's rhyme before. However, I think that it might be a variant form of the "Chickama Chickama Craney Crow" children's rhyme. A version of that rhyme is found in Thomas Talley's 1922 collection Negro Folk Rhymes, Wise And Otherwise {Kennnikat Press Edition, 1968; p. 74}. Here's that example:

Hawk And Chickens Play
(Chicken's Call)
Chickamee chickamee, cranie-crow
I went to de well to wash my toe.
W'en I came back, my chicken wus gone.
W'at time, ole Witch?
(Hawk Sponse)
"One"
(Hawk Call)
"I wants a chick"
(Chicken's Sponse
"Well, you cain't git mine".
(Hawk Call)
"I shall have a chick!"
(Chicken's Sponse)
You shan't have a chick!"
-snip-

In 1999, I created a modified version of this game which is found http://cocojams.com/games_children_play.htm In that version, children designated as "chickens" say the first verse, and the asks the child designated as the "witch" "What time is it?" The witch says an arbitrary number such as "5 o'clock". The chickens then repeat the question "What time is it? When the witch says "12 o'clock" the chickens run, trying to get to home base before the witch tags them. The last person who is tagged is the new "witch".

Of course, this game may have nothing at all to do with "I'm goin' down to smoke my pipe."