The example posted on Aug 5th ending

1 called Peter, 1 called Paul
but the third little pussy didn't have a name at all"

is another example of the folk-magpie-process.

Those lines sound as if they are stolen from the souling rhyme (N.W. England). Which used to be recited door to door on All Souls Night, before American pumpkins took over.

Soul, a soul, a soulcake
Please good missus a soul cake
.....
One for Peter, one for Paul

Though thinking about it there was also a rhyme I learnt as a child in London c1950, which at the time I never associated with the apostles

2 little dickie birds sitting on a wall
1 named Peter 1 named Paul
Fly away Peter, fly away Paul.
Come back Peter, come back Paul.

It was a finger-play. Started with 2 fists with a finger up. When the birds fly away the fingers go behind the back, then come back again.
And one for him who made us all.