Here's a link to the National Archives of Australia's INDEX for inward passenger lists. It is not complete, but includes passengers arriving by ship and by plane up to about 1966.
https://recordsearch.naa.gov.au/SearchNRetrieve/Interface/SearchScreens/PassengerSearch.aspxI wonder if your person of interest may have travelled under a different surname ... did her mum marry? if so, perhaps try under her mum's married surname...
You may find it sensible to try entering just a surname, and noticing how many entries come up showing passengers as Mrs or Miss SURNAME, no given name, many times not even an initial.
Ancestry has uploaded a number of electoral rolls up to around 1980. Have you found your person listed there? As a British born lass, assuming she migrated to Australia in the 1950s or 1960s, she would have been eligible to enrol without needing to become an Australian citizen, and she should be findable in those rolls.
There are eight jurisdictions that make up the nation, Australia. Each has its own BDM institutions, different protocols for each, and no central Australian BDM registry. Australia has strict privacy rules, and there's restricted access to non-historical BDM records, so the tree-owner is likely protecting the privacy of his living parent who is the child of your Gran's cousin.
Here's a link to RChat's Australia resources board
http://www.rootschat.com/forum/australia-resources-offers/ May I ask if you 'knew' your Gran's cousin had migrated to Australia before you actually noticed the submitted Ancestry tree?
JM