Re-reading this thread a year on stirs memories.
Although most of the songs were taught by parents as party games we played 'The big ship sails on the alley-alley-o' at infant school (W. Kensington) in the playground.
At junior school (E.Acton) games went in waves. Sometimes it would be balls against the wall, or skipping (with long or short ropes), or we used a set of steps that went all along the front of a building (10 or more could stand on the same step)to play 'days of the week', then we'd had enough of that and played 'what's the time Mr Wolf'

At Secondary school (Chiswick) there was a tradition of singing in the playground on the last day of term. I remember a big row one year because the Boys side (it was 2 schools at different ends of the building) had gone in for End of Term Service while the Girls were outside singing
She was bathing in the river and she got her *toes* all wet
But she hasn't got her - - wet
Yet
*(different each verse), tune similar to John Brown's body.

The headmistress stormed out and said there was to be 'no singing'. There was a lot of muttering and plans not to sing in our end of term service, but I think it was shortened to avoid this.