Thanks everyone.
Firstly im trying to establish if she was a British citizen at the time I was born, in order to understand if I need to apply for U.K. Citizenship or if I already am one by decent, and can simply apply for a passport. I believe if your parent was a 'British subject' at the time of your birth - according to the U.K. (there may be some discrepancy between what Australia considered a British subject to be and what the UK considered a British subject to be) then the British subject cannot pass down citizenship.
Secondly I'd like to clarify if my mother is a dual citizen, or if she gave up her citizenship when moving to Australia / when she became an Australian citizen.
It seems complicated! My mother is finding it difficult to remember / find out. Thanks!