For Ewan McVicar ten years ago, here are the lyrics to the Rushing song (my transcription from CD). It was recorded on 2 July 1941 by Count Basie and His Orchestra with Jimmy Rushing vocal and it's now widely available on CD. Apparently Vaughan Monroe also recorded it that year. The Basie band did at least five takes (see list at http://people.pwf.cam.ac.uk/djh1/lps/lp0030.html); some of them are on the French CD 'Count Basie - The Alternative Takes, Vol. 3: 1941' (see http://www.answers.com/topic/the-alternative-takes-vol-3-1941). The song was written by Don Reid and Max Chamitov in 1941; its full title is "One, two, three O'Lairy (Oh! My! Whoa! Mary!)". As you see, they took considerable licence with the old rhyme.

ONE TWO THREE O'LAIRY
(Count Basie, 1940)

One, two, three O'Lairy,
How the boys love Mary,
Cause they know when the lights are low
Oh my, whoa Mary!

One, two, three O'Lairy,
Though she's wise and wary,
Steal a kiss and steal a hug,
And then, whoa Mary!

Mary had a little lamb
But that was long ago.
Now I goes where Mary goes
No lamb could love her so.

Whoa!

One, two, three O'Lairy,
Though the boys love Mary
It's plain to see that she does love me,
Oh my, whoa Mary!

O'Lairy, two Lairy,
One Lairy, two Lairy,

Mary gonna marry me
[Ensemble]: What a bringdown!
[Spoken comment indecipherable]
Mary's gonna marry me.