Oxford Journal 27 June 1840 page 1 has obit of John Vickery who died suddenly on 18th June. he had been governor of Clerkenwell House of correction. It mentions that he had a son with an official position in New South Wales.
Ancestry has his burial record, buried on 20th. his age at death makes his year of birth about 1772.
This could fit with being Christopher and Joshua's father.
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There's John Vickery's will on ancestry, mentions Bow street officers, so same chap who died suddenly in June. Also mentions sons John, Joshua and Christopher, daughter Phoebe, brother James, son in law Joshua Armstrong. rather difficult to read.
more added.
After names of sons Joshua and Christopher it says Vickery otherwise Brown

And another son James George Vickery otherwise Brown.
It sounds as if they might be stepsons and Brown was their original surname.
then this sentence:
being my three .... or reputed children by my late ....Ann Brown deceased.