Joybell, I have always pronounced 'cherry' to rhyme with chair-ie. And I'm not Aussie.

Alice Gomme, "Traditional Games of England, Scotland and Ireland," part 2, pp. 25-35, has 26 versions, but no cherries.
Halliwell (1846, clxix) has "the song on the bells of Derby on football morning" (Found no Derry):

Pancake and fritters,
Say All Saints and St. Peter's;
When the ball come,
Say the bells of St. Alkmum;
At two they will throw,
Says Saint Werabo;
O! very well,
Says little Michel.

Fritters - St. Peter's?

Gomme goes on to speculate: "Oranges and lemons were, in all probability, originally meant t mean the colours of the two contesting parties, and not the fruits of their names. [I don't think that likely]

Gomme doesn't have much to say about the 'chop - chop,' but it appears in several rhymes:

Here comes a candle to light you to bed,
Here comes a chop'n bill to chop off your head- Chop- chop- chop- chop.
Another:
Here come some great candles
To light you to bed,
Here come some great choppers
To chop off your head.
etc., etc.

The possible relationship to the humorous "Benjamin Backstay" is interesting. Dibdin's songs were widespread in the States and Ccanada, and one would presume Australia as well. They were sold widely on penny songsheets, many without attribution to Dibdin.

Coincidentally??- a Benjamin Backstay, a burglar, age 20, was sentenced to death at the Old Bailey in 1827.