The Australian Constitution (heavily based on the US, as the UK has never had a written and entrenched constitution and is not a federation as the Commonwealth of Australia and the United States ARE) only mentions 'god' in the preamble which is not specifically part of the Constitution but is part of the covering clauses. It is well known amongst those who have studied these things that it is ONLY there as a trade-off to the token religious whacko on the drafting committee, so as to allow the inclusion of s116 which dis-establishes religion.
Also its important to note the differences so that you don't quote US rights when they don't apply in Australia. Such as free speech, which is worded differently, more in the right to hold an opinion. Also quite interesting that Australia chose to opt out of banning domestic use of racial hate speech and propaganda with regards to rights, although there are laws that cover hate speech.
Federation was mostly pushed for by businesses wanting to conduct cross border trade, as pre-1901 you needed to pass border control to pass between colonies such as NSW and Vic, including tariffs and everything you'd see between two different nations. The idea it was something done for the greater population alone is misleading. Victoria and Melbourne in particular had become wealthy and also xenophobic, they didn't want foreigners but they wanted trade to increase wealth and luxury goods especially from within Australia being of mostly all the same ancestry. The businesses also played on fears of invasion by nations such as Russia/France/Germany post Crimean war as Australia was barely protected by itinerant RN Vessels and had no significant navy or army of its own. So the reasoning behind Federation in Australia and Independence of the USA is a vastly different story with a very different set of 'founding fathers'.
It's sad that most Australians have more idea about the US system and history then what happened in the creation of Australia. Especially in the lead up to Federation, everyone knows there was a gold rush, but very few know the story and lessons to be learnt from what happened in Melbourne post gold rush as well. There are some good historical lessons about shunning the world and thinking you can live off limited wealth for ever.