A favourite from when I was 5 (it seemed sooo rude then!)

Mary had a little lamb, she thought him rather silly
She threw him up into the ari and caught him by his

Willy was a watchdog sitting in the grass
Along came a bumble bee and stung him on the

Ask no questions tell no lies
Ever see a policeman doing up his

Flies are a nuisance, bees are worse
That's the end of my silly little verse

I tried searching for some of the lines and came up with
online dictionary of playground slang

A choosing game like some of the ones mention above went:

ippy dippy dation
my operation
how many platforms at the station
1,2,3 .. (etc stop when you get to someone you want to be "it", in your football team etc)

Some years back there was a project in Camden (that's in London UK) which collected children's songs from around the world all rfom children in schools in the borough: Camden World Song project