Australia would be offering help at the earliest possible opportunity once the war was acknowledged, though probably to Scrimgeour not the Order. I would guess they'd send reinforcements to the Aurors or some sort of specialised help (magical tracking using indiginous techniques not used in Britain is something quite possible). You are going to have to do something for Anzac Day. But I would guess that during the colonial period and right through WW1, there wouldn't be the separation between the magical and muggle European settlers that there would have been in Britain and would be in Australia now. There'd be too few people, life would be too hard and Australian society was based upon co-operation to a very large degree (more so than in the USA during their early days).