I only heard Inky Pinky Ponky (which I posted before I went off for the long weekend) once, it was chanted while the teacher's back was turned by the girl in the next desk.

We put the "chopper to chop off your head" after Oranges and Lemons.
Played as described, but when a child was caught the two making the arch would ask "Oranges or lemons?" and you would whisper your choice. The arch people would have decided which of them headed the Oranges and which the Lemons, you would be told who to stand behind. When everyone had been caught the game ended with a tug-of-war.
This was a game we were organised into playing at childrens parties (up to age 10 I suppose, though there would be younger and older cousins and siblings).

Have we mentioned
Solomon Grundy born on Monday.....Died on Saturday, Buried on Sunday
That was the end of Solomon Grundy.
And
Old Roger is dead.

Those were two more that I only heard from adults not other children.