Circa. 1956/7 I sang this song for my grandfather when he came to visit my father, I sang the version that I had learned from my siblings and from the street skipping, (I was reared in a small suburban housing-estate in northside Dublin) my grandfather said that he also learned it from the skipping.
It was mostly girls who would skip they would use a shortrope, two would swing the rope and a third would run in and skip, they had all sorts of songs and rhymes to keep time. If there was less than three girls then a young boy might be pressed into turning one end of the rope and to help sing out the rythyms.
My grandfather said that he had learned the proper words and he then proceed to teach me the variant of the song that he learned. This is how I remember it.

River Sáile

There was an old woman who lived in the woods
we la, we la, wall la,
and that old woman she wasn't very good
Down by the river Sáile

She had a baby three months old
w
and that little babby was very bold
D

She had a penknife long and sharp
w
She stuck the dagger in the babby's heart
D

She stuck the penknife in the baby's head
w
The more she struck the more it bled
D

She buried the baby in the wood
W
The neighbours they all saw the blood
d

Three hard knocks came knocking on the door
w
And the woman fell down in a faint on the floor
d

T'was two police men and a man
w
and another ouside waiting in the van
d

"Are you the woman that killed the child?"
w
She said "I am" and they went wild
d

They took her away and they put her in the jail
w
Loudly she did bawl and loudly she did wail
d

They put a rope around her neck
w
And dragger her up onto the deck
d

The rope was pulled and she got hung
w
Round and round her body swung
d

Now that was the end of the woman in the woods
w
And that was the end of the babby too
d

The moral of the story is
w
Don't stick a penknive in a babby's head
d

My grand father had me sing the first line, then he would reply in answer with the second line. He said that was how he learned it when he was a child, he was born in the heart of Dublin city in about 1890. He died in 1974 (Ar dheis Dé go raibh a ainm).
You seldom see street skipping in Dublin anymore, and you hear precious little of "the haunting childrens rhymes, that once were part of Dublin in the rare ould times"
Fergus