Azizi, you quote the Opies saying that
My boy friend's name is Fatty,
He comes from Senoratti
With turned up toes and a pimple on his nose,
And this is how the story goes:
came to the UK in 1959.

It seems to have some relation to the 2 ball rhyme we said in 1954 (I had just started secondary school in London, age 11)
You start with the letter A and all the names have to begin with that letter
e.g.
My name is Alice
My husbands name is Arthur
We live in Aldershot
And we sell apples

You go through the alphabet while throwing and catching the balls, if you cant think of something or you drop the balls it's the next ones turn, and when they are out you go start again with the verse you were out on.
Our version had rythmn but didn't have to rhyme.