The answer to that question, as far as we can tell, seems to be no.
In a thread posted over at the official Respawn website, one user posted an email chain with Abbie Heppe, the Community Manager at Respawn, who stated that it was "really a question for Microsoft" before writing, in a later email, that the servers used during testing of Titanfall were not Australian. Most likely they were based in Singapore.
So will Titanfall have local Australian servers? The answer is most likely no.
But Abbie Heppe is correct -- this really is a question for Microsoft. It's generally believed that any local servers used for Xbox One games in Australia will be run from its Azure Centres, which are dotted around the globe. Microsoft has already announced plans to build two local Azure Centres in Australia but hasn't announced much beyond that. These centres will be located somewhere in Victoria and NSW, but we don't know precisely when they will be established. Microsoft has stated its Azure Centres will launch later this year, but it's extremely doubtful they'll be up and running for the launch of Titanfall.
In addition, these centres are primarily aimed at business users who want to set up their own cloud based infrastructures without setting up their own data centres, but the expectation is they'll also be used to more front facing services like local servers for games like Titanfall -- but even that hasn't been confirmed at this stage.