Some male singer [Raffi?] has recorded this version of that song. Here are the words to that song that I can remember from memory, probably with words changed and the order too [the folk process at work??!]

There once was a man and he was mad
so he jumped into a pudding bag.
The puddin bag it was so thick
that he jumped into a walking stick.
The walking stick it was so narrow
he jumped into a wheelbarrow.
The wheelbarrow began to break
so he jumped into a chocolate cake.
The chocolate cake it was sour
So he jumped into a case of flour.
The case of flour caught on fire
and blew him up to Jeremiah. Whew!

end of lyrics

This type of rhyme where one defective item is exchanged for another one that also proves to be defected reminds me of the "Went To The River [but couldn't get across] songs composed by enslaved African Americans...

"Gray and Black Horses' is one song that contains a [now standard] series of exchanges:

I went to de woods an' I couldn' go 'cross.
So I paid five dollars fer an ole gray hoss.
De hoss wouldn' pull so I sol it for a bull.
De bull wouldn't holler, so I sol it for a dollar.
De dollar wouldn't pass, so I throwed it id de grass.
Den de grass wouldn't grow. Heigho! Heigho!
   {Source Thomas W. Talley "Negro Folk Rhymes", 1922]

"Apples On A Stick" {Barbara Michels, Bettye White}, a 1983 book of African American children;s rhymes collected in Houston Texas has this version of that old song:

Shake shake shake
Eeny Meeny
That's a queeny
Ooh ba Thumbelina *
Ah cha ca che Libarace
Oh baby I love you
yes I do
Take a peach
Take a plum
Take a piece of bubble gum
No peach
No plum
Just a piece of bubble gum
Ooshe ahshe
Ooshe ahshe
I want a piece of pie
The pie too sweet
I want a piece of meat
The meat too tough
I wanna ride the bus
The bus too full
I wanna ride the bull
The bull too black
I want my money back
The money too green
I want a diamond ring
end of quote

(There is also an alternative ending for this series of strung together rhymes: the money too green/I want a jelly bean/the jelly bean too white/it's time to say good nite.. or as one person wrote it
on the Wheee! Blog school games thread:

"the jelly bean to white
goodnight sleep tight, don't let the bed bugs bite
if they do get a shoe an beat the black and blue"

more children's rhymes here

*From my childhoold in the 1950s Atlantic City, New Jersey
I recall the "eeny meeny parts of this rhyme, but the line went
"Ooh Ah Thumbelini". We then chanted "Ahche mache cucharachi/I love you" and ended the rhyme there. I thnk we just chanted this rhyme or used it as a counting out/elimination rhyme.

The longer version of 'Take a Peach' might be a handclap or jump rope rhyme..

Does anyone have any information as to whether these type of trade songs are found in other traditions than African American?

Ms. Azizi