Sure can.
I waited until I had my hysto before I changed my gender, as I was able to get my hysto covered by the government while I was still legally female.
There were mere months before I “had” to get my gender changed, in order to be able to legally marry my Wife. We’d booked out wedding, sent out invitations etc.
Right after my Hysto I began the process of getting my marker changed on my birth certificate. There were forms I could download from the Births, Deaths and Marriages website, which included explicit instructions detailing what evidence I must provide etc.
The “main” thing was the form requiring two independant statutory declerations (legally binding) from two different Medical Doctors which had either administered Hormone Treatment or Surgery pertaining to my gender reassignment.
My chest surgeon was a few hours drive away, and ridiculously expensive for even a consult requiring just a signature. I decided to get my treating GP (of almost 8 years at that stage) to sign one of the stat. dec’s, and my Hysto surgeon the other. It was pretty specific in the fact that they both had to be present and witness each other’s signings, but luckily my GP was super accomodating and met me at the Hysto surgeon’s consult rooms one day after they’d both finished work.
I was super paranoid that B, D & M’s wouldn’t, for whatever reason, accept my paperwork – so I overcompensated by providing official copies (signed by a Justice of the Peace) of every single piece of medical paperwork, psychological reports etc pertaining to my transition to date. I also had to pay a fee somewhere in the vicinity of $60-$100. I can’t remember exactly.
All in all, I certainly went overboard but the turn around time was fairly rapid for a Government organisation, and I had my ammended Birth Certificate in the mail. Previously my old Birth Cert had, on the reverse, a “change of name” declaration, indicating my name had been changed from BIRTH NAME to CURRENT NAME in giant, bold letters. Now, there is no such thing on my Birth Certificate – it says “Male” next to my Sex, and has my current name without mention of previous – which was a welcome relief.