I remember seeing a 'Jody' drill team at Fort Sam Houston in the 1940s. I don't remember if it was called that, but I think it was. The earliest reference to Jody that I can find is from 1944: A. Lyon, "Unknown Station 28
Ain't no use in going home,
Joady's got your girl and gone.
Gonna get a three-day pass,
Just to kick old Joady's ass."
Subsequent references (Beecher, i945, etc.), spell it Jody.
From J. E. Lighter, "Historical Dictionary of American Slang," vol. 2. Jody is the mythical character who takes care of the wives of men away on military service. Himes, 1945 in "If he hollers," spoke disparagingly of a "black jodie in his fine jodie clothes."

The team we saw was not from there, but was brought in from another base, possibly Ft. Benning. It seems to me (a long time ago, my memory is uncertain)) that this black drill team performed all over the country. At the time, the forces were not integrated. (Tried on google; nothing before 1950 found).

The History Channel.com has some Jody cadences: http://boards.historychannel.com/threadedout.jsp?forum=89&thread=300018494

It does seem likely that the "Mama" cadence chants came from a children's rhyme, but no firm evidence.
Here is a "Mama, mama, can't you see" from the USMC; submitted by HM3 Rembert, ER:
Mama, Mama, can't you see,
What the Corps has done to me,
I used to drive a Cadillac,
Now I wear a cammy pack.

Mama, mama, can't you see,
What the Corps has done to me,
I used to go outdoors to play,
Now I'm humpin' every day.

Mama, mama, can't you see
What the Corps has done to me,
I used to ride my cousin's hog,
And now, I'm just a DEVIL DOG.

Mama, mama, can't you see,
What the Corps has done to me,
Cammies pressed and boots are clean,
Cause I'm a U. S. MARINE!

Put me in a barber chair,
And they shaved off all my hair.

Put me in a silver jet,
The ground was cold, the air was wet.

Took away my faded jeans
Now I'm wearing cammie green.

This and more jodies at Special Operations.com: Jodies
Click on USMS and USAF, the USA and USN sites are under construction.

Here is an advisory about a 'run' at Fort Bliss; cadences permitted, but they must be from the "Air Defense Artillery Jody Cadence Book."
The jodies seem to be alive and well. Fort Bliss Post Run

I couldn't find anything on the history of Jody teams, WW2. Perhaps in these volumes:

Johnson, Sandee Shaffer, ed: Cadences: The Jody Call Book No. 1, Canton, Ohio, Daring 1983, 152 pp.
---"--- Cadences: The Jody call Book No. 2, Canton, Ohio, Daring, 1986, 153 pp.
The US Army Infantry Center, Ft. Benning, GA, has a booklet: Jody Calls, 1984, 46 pp.