I think this is a related (possibly Irish) song to "The Cruel Mother" (Child #20). Often as a jump-rope rhyme or some other game song. Dominic Behan (on Streets of Songs) and the Clancy Brothers (on The Clancy Brothers and Tommy Makem in Ireland) sang this. Unfortunately I don't have these LPs.
(From: HERE)

WEELA WALLIA
(Probably The Clancy Brothers' version)

There was an old woman who lived in the woods
Weela weela wallia
There was an old woman who lived in the woods
Down by the river Sallia

She had a baby six months old
Weela weela wallia
She had a baby six months old
Down by the river Sallia

She had a penknife three foot long
Weela weela wallia
She had a penknife three foot long
Down by the river Sallia

She stuck the penknife in the baby's head
Weela weela walia
She stuck the penknife in the baby's head
Down by the river Sallia

Three big knocks came a-knocking at the door
Weela weela wallia
Two policemaen and a man
Down by the river Sallia

"Are you the woman what killed the child?"
Weela weela wallia
"Are you the woman what killed the child?"
Down by the river Sallia

"I am the woman what killed the child"
Weela weela wallia
"I am the woman what killed the child"
Down by the river Sallia

The rope got chucked and she got hung
Weela weela wallia
The rope got chucked and she got hung
Down by the river Sallia

The moral of this story is
Weela weela wallia
Don't stick knives in babies' heads
Down by the river Sallia

~Masato