What is your visa status? This will determine if your child is eligible for British citizenship. If you have indefinite leave to remain ILR, then your child is a British citizen. If you are still under visa restrictions (eg work, study, ancestory) then your child is not a British citizen.
It may be eligible for Australian citizenship, depending on how parents acquired their citizenship (eg birth, naturalization, or descent), but it is a 2 step faffy process to get an Australian passport for children of Australian citizens born in the UK. Get on to it ASAP so the baby will have at least one passport, particularly if you didn't get your own citizenship through birth.
If you don't have ILR, your child will be eligible for British citizenship when you are granted ILR, so the child will be eligible to get their British citizenship12 months before you.
There is no problem holding both Australian and British citizenship, Australia has allowed dual citizenship since 2004, and the UK has never had an issue with dual citizenship.
Just to let you know, if you go back to oz, there are some jobs in oz that ban dual citizens, including ASIO, lots of DFAT stuff, national security jobs, and to become a member of parliament.
starting point for researching British citizenship
www.ukba.homeoffice.gov.uk/britishcitizenship/eligibility/
starting point for researching Australian citizenship for babies born overseas
www.citizenship.gov.au/
ask me any questions you want on this, I have been juggling visas and citizenships for years.