Hi, thanks for your question.
In the event of a successful referendum in Australia there is no requirement for the British Parliament to be notified. Australia has passed laws which give it Constitutional independence from the British Parliament (most recently the Australia Act 1986). This means the Australian Parliament can pass bills, including successful referendum bills without notifying the British Parliament. The process for conducting a referendum is contained in Section 128 of the Australian Constitution, and it does not include any requirement to notify the British Parliament.
However, Section 128 of the Constitution does state that if a bill is agreed to at a referendum, it must still be presented to the Governor-General as the Queen’s representative in Australia. Because the Governor-General is a part of the process they would notify the Queen about the successful referendum through official communications.