Here's how it's rendered at http://www.kididdles.com/. No songwriter attribution at Kididdles on this one. ASCAP says "Tinga Layo" was copyrighted by songwriters Charity Bailey and Eunice Holsaert: and "Run Little Donkey" by Eugene Raskin and Louis E. Gottlieb (was it a Limeliters recording?). BMI lists songs titled "Tingalayo" by Peter Allen Lurye & William James Cobin, by Peter & doris Kaplan, and by Bob Singleton; and "Tinga Layo" by C. Barny Robertson. I think we can call it "Traditional" and let all these people fight about it.
The Fireside Book of Fun and Game Songs does not name the songwriter, but says it was copyrighted in 1943 by M. Baron Company. The Fireside book says it is a calypso song from the West Indies. I sing it "Tingalayo, run little donkey run." I don't sing the peanut butter verse. I have some self respect, you know....
-Joe Offer-

Tingalayo

Tingalayo, run my little donkey run
Tingalayo, run my little donkey run

My donkey hee, my donkey haw
My donkey sleep in a bed of straw
My donkey short, my donkey wide
Don't get too close to his backside

Tingalayo, run my little donkey run
Tingalayo, run my little donkey run

My donkey walk, my donkey talk
My donkey eat with a knife and fork
My donkey eat, my donkey sleep
Don't get too close to his hind feet

Tingalayo, run my little donkey run
Tingalayo, run my little donkey run

My donkey laugh, my donkey cry
My donkey loves peanut butter pie
My donkey low, my donkey high
My donkey loves peanut butter pie

Tingalayo, run my little donkey run
Tingalayo, run my little donkey run
Tingalayo, run my little donkey run
Tingalayo, run my little donkey run

^^