Brian, Thomas Plumer Addison arrived in 1838 in Adelaide on the Pestonjee Bomerjee with his family who settled and remained in Adelaide. Prior to coming to Australia he lived in Holborn, London. His wife was Eliza Mary Claxton Tourner. They had 7 children (3 in England, 1 in transit, 3 in Australia). I believe Thomas had a brother Robert Addison. They appear to have been in business together in London prior to Thomas emigrating to Australia. I also believe that Thomas had a son named after his brother - Thomas Plummer Addison.
I do not know if there were other Addisons in Australia prior to TP but it would seem likely as the name is not uncommon. I have more information regarding TP' descendants in Australia if you are interested.