Chicago, early 1940s:

One, two, three a'leary,
Four, five, six a'leary,
Seven, eight, nine a'leary,
Ten a'leary, USA! (Patriotic WWII times, those)

U.S. Midwest, 1940s:

One, two, three a'leary,
Four, five, six a'leary,
Seven, eight, nine a'leary,
Ten a'leary, POSTMAN! (unexplained, until we heard Dominic Behan sing, in London, 1958:

Open the door and let me in sir,
I am wet unto the skin, sir.
Open the door and let me in, sir,
All to post me letter.
One, two three a'lairy,
Four, five, six a'lairy,
Seven, eight, nine a'lairy,
Ten a'lairy, POSTMAN! (Dublin, ca. 1930s)

Vermont, early 1960s:

One, two, three o'leary,
My first name is Mary.
If you think it necessary,
Look it up in the dictionary.

Boston, early 1960s:

One, two, three o'lairy,
My first name is Mary.
Don't you think that I'd look cute
In my father's union suit? (That's how we heard it, Barry, from a woman who would be a bit older than your sisters could have been.)

And, finally, Jeannie Robertson, in Aberdeen, Scotland, in 1958, sang for us:

One, two, three a'leery,
I saw Wallace Beery,
Sittin' on his bumble-eery,
Kissin' Shirley Temple.

Actually, we have found one more, but I can't think of it right now. Something about "Mistress Mary looking like a chocolate fairy..." Perhaps on her bumble-eery, too. I dunno. Maybe another Mudcatter can help.

By the way, Jean Redpath pointed out, as she was doing one of these ball-bouncing/leg-cross-over games, that there is a line in Piers Plowman about beggars sitting at the city gate, "Legs aleery." I'm not sure of the spelling, but the meaning was clear: they had their legs crossed, i.e.: sitting in what we call "Indian fashion." In the game, of course, the player crosses her leg (this is usually considered a girl's game) over the bouncing ball.

Hope this helps.

Sandy