Some states have a statewide system, for example NSW and their Complying Development Code (CDC). Other states manage it at a Council level, and you may find differences between Councils as a result. One common feature, though, is that the process of verifying the project ticks all the necessary boxes for approval is taken outside of Council to an external consultant. This is usually a private Town Planner, but may be a Building Surveyor. By doing this, you bypass any backlogs in Council, as you have a lot more options to choose from.
A typical timeframe for a fast track development is just a few weeks. That's not set in concrete, but unless you're wrong in thinking your development complies, most of the time it applies.
Let's take a look at how it works, state by state....Queensland
In Queensland the process is known as Risk Smart, and is managed at Council level. The main Councils in South East Queensland use it, including the Brisbane, Logan and Gold Coast City Councils. Outside of those areas, check with your local council.New South Wales
NSW has the Complying Development Code (CDC) which applies statewide. Here's a good place to start https://pp.planningportal.nsw.gov.au/onlinecdc