I can find a Japanese-style hot spring near Sydney. In the Blue Mountains, there's Japanese Bath House, which according to visitnsw is "the only Japanese onsen (hot spring) in New South Wales where you can experience the Japanese spa culture."
However, are there any non-Japanese hot springs within day-trip distance of Sydney? Any non-Japanese style is ok, so for example a Turkish-style hot spring would be ok. Australian-style, if there's such a thing, would be an added bonus.