Hi DBB,
I'm confident that pjf, with whom I've been exchanging emails, has found the "right" William Brunton in Australia, because of his unique name "William Tait Brunton". His full name is on Australian records.
He was born 30 Sep 1831 in Yarrow, and married Agnes Orr on 6 Jun 1856. They had children Margarate Kennedy (18 Mar 1858), Janet (30 May 1860), David (1864), George (30 Jun 1866), James Kennedy (1868), Robert John (21 Oct 1870), Agnes (1873), and William (1874).
I have pecilled in his parents as David Brunton (5 Apr 1802, Yarrow) and Janet Scott (1801, Selkirk), m. 14 Nov 1824, Yarrow. But I could well be wrong about that (there were several Bruntons from Yarrow), and I see you have them producing another William.