I'd guess Azizis rhyme has a separate origin - it fits in with the tradition of songs in which childcarers sing about the terrible things that might happen to a baby.
(eg Way down yonder, down in the meadow,
There's a poor little lamby,
The bees and the butterflies peckin' out its eyes,
The poor little thing cried mammy.)

I remember somewhere that gorn dahn the plug'ole is supposed to be based on a true incident of something that happened in a communal washing facility where they had extra big plugholes. Which is a rather disturbing thought.   But perhaps it was just something that nearly happened, or that the audience would recognise as something that might almost happen:

Here is a slightly extended version, as recorded by Ginger Baker (of Cream):

A mother was bathin' her baby one night
The youngest of ten, a poor little mite
The mother was fat and the baby was fin
T'was nowt but a skellington wrapped up in skin

The mother turned round for the soap from the rack
She weren't gone a minute, but when she got back
Her baby had gone, and in anguish she cried
"Oh, where is my baby?", and the angels replied

Your baby has gorn dahn the plug'ole
Your baby has gorn dahn the plug
The poor little thing was so skinny and thin
He shoulda been bathed in a jug

Your baby is perfik'ly happy
He won't need no bathin' no more
He's workin' his way through the sewers
Not lost, just gone on before

Your baby has gorn dahn the drainpipe
And the chlorine is bad for his eyes
He's havin' a swim, and it's healthy for him
He needed the exercise

Don't worry 'baht 'im, just be 'appy
For I know he is suff'rin' no pain
Your baby has gorn dahn the plug'ole
Let's hope he don't stop up the drain

ALTERNATE VERSE:

Your baby is perfik'ly 'appy
He won't need a bath any more
He's muckin' abaht with the angels above
Not lost but gone before