One version I know sings very like yours Q

The ants go marching one by one hurrah, hurrah
The ants go marching one by one hurrah, hurrah
The ants go marching one by one,
The little one stopped to (insert appropriate noun and verb here!)
And they all went marching across the floor, under the door
out in the rain, down the drain
ZOOM, ZOOM, ZOOM

That is more modern though. I first heard it in the 80's, but the person I heard it from had had it from her mother, so 60's perhaps?