Numerous nuclear weapons tests were conducted on the Australian continent, including Operation Hurricane, the test of the United Kingdom's first atomic bomb on the Monte Balle Island archipelago, in 1952 followed by two more tests in the same location as part of Operation Mosaic in 1956.
So why didn't Australia ever take inspiration from it's allies and begin acquiring nuclear weapons of it's own to protect itself?
The benefits of a nuclear arsenal are shared across the alliance. The downsides you'll have to carry yourself.
Why would Australia ever have a motive to desire a nuclear arsenal? There are only ten countries that developed nuclear arsenals: The five official ones (US/UK/FR/RU/CH), the three illegal ones (India, Pakistan, North Korea), the presumed illegal one (Israel) and the former illegal one (South Africa).
The United States developed nuclear weapons to win World War 2 in 1945 and to strengthen its global hegemonial claim, especially in the already visible postwar world.
The Soviet Union developed nuclear weapons to counter the American development, which would have made World War 3 far more likely and far more dangerous to the Soviets if the USSR had not counteracted the American program with its own in 1949.
The UK developed nuclear weapons practically alongside the U.S. - although the deployment of their first own device would take until 1952. As the U.S., especially in the person of Harry Truman, grew more vary of having authority over American nuclear devices in anyone's hands but the president's, the UK saw itself coerced to drive off into its own nuclear program. The third country heavily involved in the Manhattan Project by the way was Canada. Canada abstained from a nuclear program for the same reason Australia has - there is absolutely no upside in having one. The UK in the early 1950s still had delusions about being one of the three superpowers - the model would ultimately fall down to a system of only two superpowers after the 1956 Suez Crisis had shown the world that Britain was simply no longer able to compete.
France in 1960 finished the bomb as a direct result of the aforementioned Suez Crisis. The French mentality after World War 2 had been one of deeply offended national pride and the defeats in 1940, 1954 and 1956 - the last one with a perceieed betrayal by the U.S. and UK - led the French to seek their own path away from the Angloamerican power bloc, closer aligned to BeNeLux, West Germany and Italy. This self-imposed isolation from the U.S. and the UK forced France in its own perception to develop nuclear capabilities of its own.
China in 1964 had similar motives than France - China too had taken a step away from its superpower buddy, in China's case the Soviets. After the Sino-Soviet split, the relaxation of totalitarianism in the Soviet Union immediately threatened to put into question the Maoist style of governance and pushed China into a third position between the United States and the Soviet Union.
India and Pakistan developed nuclear weapons primarily to counteract each other. Both countries see each other as economically important to third parties and thus cannot rely on foreign nuclear arsenals to resolve the ongoing conflict between the two. India was the first to deploy a device in 1974, Pakistan immediately developed a nuclear program of its own, although its first test only took place in 1998.
North Korea is, like India and Pakistan, isolated. China is its protective buddy, sure, but can China really be trusted; with Chinese wealth and prosperity so dependent on openness and friendly trade relations with the west?
Israel and South Africa likely developed their nuclear weaponry in secrecy and in cooperation. The "Vela Incident" of 1979 in the ocean south of the Cape of Good Hope was likely a joint effort by the two. Israel needs nuclear weapon as an immediate deterrent - unable and unwilling to wait for American assurances and U.S. nukes in the case of military invasion by or into its neighbors. South Africa, which has since the end of Apartheid dismantled its arsenal, was another case of internationally isolated authoritarian state.
So, does Australia have any of these?
A world war to win
An annoying neighbor to threaten
A hostile (super-)power to match in power
Hurt national pride to heal
Its own superpower status to prove
Imminent, sudden and high-pace invasion to deter
Foreign threats to convince of not attacking its authoritarian, downtrodden, internationally isolated regime
Didn't think so. Australia is allied to plenty of nuclear powers. Why waste your own money on nukes when the Americans, Brits and Frenchies do it for you? The UK and USA protect Australia from any sort of nuclear threat with their own nuclear deterrence, and that comes free of unnecessary costs - unlike a decades-long nuclear program. There is nothing Australia needs to "protect itself" from. It has no unique enemy or adversary.