While Australian users of Twitter can send updates via SMS if they're prepared to pay for a text to the UK, there's currently no option for receiving updates the same way. However, in a post discussing a recent deal allowing Twitter updates to be by Canadian users, Twitter also promises that Australia is in the running for a similar service:
Stay tuned for more support all over the globe as we make progress. And fear not Australia—you're on the list too!
How long it will take is anyone's guess. No doubt the big issue is that Aussies receive texts at no charge, unlike US plans where the recipient typically pays (so if a deal does go ahead, it will almost certainly be single-network and very possibly tied to a particular plan).