I'm an adult and was diagnosed 4 years ago aged 31 in NSW. I've since moved to SA.
As an adult, you CANNOT be assessed or prescribed ADHD stimulant medicine formally (and therefore diagnosed officially) by anyone other than a Psychiatrist. To complicate things, only very limited numbers of Psychiatrists will agree to treat Adult ADHD. I have plenty of opinions why but they don't matter. The point is, you have to find one who will even see you!
There are two options here. You can ask if your GP knows any of them. If not, then I'm afraid you will need to call around them all to ask.
This may be difficult as specialists treating it in adults are very very limited - not a lot will.
The process is something like this:
Expect a waiting period, perhaps of months, before you go and see them for assesment. I believe NSW is now like SA in that once diagnosed and treatment arranged, an authority can be given to a GP so that they can prescribe you the medicine for a period of 12 months, when you are reviewed by the Psych and re-authorised.
As for finding one nearby:
Well, this all depends on where in NSW you are. Due to the low number of treating Psychiatrists you may need to travel significantly (for the assessment/treatment stuff initially, then if transferred to GP, once a year after that).
You can search online ot see if a list is available, however expect it to be out of date or inaccurate. The list I found for my new location was only 1/3 accurate.
Best of luck with finding someone to assess you. If you're diagnosed with this condition best luck in managing it well. If you get an appointment take along self reporting scales AND reporting scales that you give to others - like your parents, anyone you live with, etc. They are useful for the doctor to have since he can't ask about how you presented, ESPECIALLY the parent one. I had my mum do one for me.
Finally - with the bipolar diagnosis it may be harder (I do know people with both, and it is common enough). Some symptom overlap is possible.