Remembered from East London in the 50's:

Chase me Charlie, Chase me Charlie,
Chase the leg of me drawers
Chase me Charlie, Chase me Charlie,
What's the size a yours?

(Two balls rhyme)One two three o'lairy,
My ball's up in the airey,
Don't forget to give it to Mary,
NOT to Charlie Chaplin

I never got to hear any more of the 'Auntie Mary had a canary up the leg of her drawers', as the adults muttered the rest when the kids were around.
Strangely, they didn't mind singing:
'You ought to see my granny by the fire - toasting her crumpet.
All day long she sits by the fire, toasting her crumpet.
And as she sits she sews, and as she sews she sits,
You ought to see my granny by the fire, toasting her crumpet.

another kid's nonsense rhyme I remember:
My name is ell-i, ell-i, chickal-i, chickal-i, ompompiley, ompompiley, willie willie whiskers, willie willie whiskers,
cowboys and indians, BIG CHIEF (said with two fingers in a V sign, knuckles to forehead.)