See the Uniform pages for variations of these.
Australian Red Cross Society.
Tasmania Police force
NSW or Victorian Artillery
If you look at Cossum’s book, the NSW Artillery on page 8 and the Victorian Artillery Staff on page 16 are the same, apart from the quoted manufacturers. This is because the design was copied directly from the British, unlike, for example the Royal Victorian Volunteer Artillery that had their own design struck.
The backmark relates to the wholesalers of these buttons. Barron & Wilson were cutlers/ironmongers in London from at least 1824 until 1885, when it became Wilson & Son at King Williams Street, London. They were sporting goods/kitchen/hardware merchants. Therefore the button dates post 1884.
If this dating for Wilson is correct, then the button was for the Victorian Artillery, as Cossum dates this design as 1870-80 for New South Wales.