Hello,
We are of a similar situation; recently had a baby and are heading back to UK to live.
Because I am a UK Citizen, our baby's entitled to a UK Passport, even though she was born (and her birth registered) in Oz.
So we have just applied to the British Embassy in Canberra for her Passport.
UK in Australia - the British High Commission in Australia
Maybe you could do the same if you are UK Citizen, then your baby won't need a Visa.
Good luck with bubs!
Jo x