Hi there, Welcome to RootsChat
Excellent professional help available in Australia, various organisations, including
http://www.jigsaw.org.au/ Some thoughts ... Seems likely your Mum was born in the UK, and perhaps came to Australia after adoption... Have you checked the National Archives of Australia who may well have immigration papers ... not just the passenger list, but more detailed papers, perhaps not readily available to the general public via online digitised displays. I mention this because perhaps ten or so years ago, my brother in law and his wife found their full folder of applications, medical hisitory/certificates, from their local doctor, dental records etc from when they applied to come to Australia (ten pound poms scheme, plus the same for each of their then children, they arrived 1960s) ... all digitised, their signatures, their then address in England, comments from their immigration interviews, not just the ship passenger manifest, and much more. Not long after, the NAA realised Privacy issues were involved, and the entire suite of papers were no longer online. BUT the papers do exist at the archives.
http://www.naa.gov.au/ http://www.naa.gov.au/about-us/contact/JM