Information from Sue Woolf ne Barnard is as follows:

We are a very old Jewish family in England since the 18th century or even earlier, with some branches in Australia. But that is a diversion because the connection is through the Joshua
family. I suspect that they came from Germany but were certainly in England
by the 1760s.

You will be wondering what this has to do with your family. Michael's
eldest daughter Betsy married Charles Jacobs in May 1846, the first Jewish
wedding in Adelaide. They had numerous children of whom the 6th was John
Jacobs who married Kate Cohen. Incidentally their daughter Beryl Jacobs
married Louis Alkin Gollin, not Loris Gollen; their son Keith Jacobs
married Kitty Goldstein in Sydney in 1918. The source for Betsy & Charles
was a copy of their ketuba and a book by John S Levi called" Australian
Genesis". The info about Beryl and Keith was from a great grandson of
Betsy and Charles when I met him last year.
Michael's 2nd daughter was Esther Joshua who married Solomon Lyon Marks in
1850 - I've yet to discover whether that was in Adelaide or Melbourne. At
some stage they changed their name to Leicester, or as written on a fairly
illegible snippet of family tree Esther Lestor. One of their sons was
Herbert Leicester who married Amy Frankel. I'd be interested to know where
you acquired the information on Herbert and Amy's descendants as this was
the part of my family on which I had the least information - only a tree on
the internet some 2 or 3 years ago. Many of them still live in New
Zealand, particularly the Sheats, and my NZ cousin would like to make
contact.
The 3rd daughter married a Barnett in London, and after a few years in
Melbourne settled in New Zealand. I am in touch with one of their
descendants.
The fourth daughter was Rachel Joshua, my great-grandmother.