I wish that I could help but, in my 6 years living in Darwin (1993 - 1999) I never met any woodturner nor I seen or heard of any shop dedicated to this art so, and unless things have changed since I left, you aren't going to bring home any wood from there, unless someone reading this and living in Darwin or in any of the big outer suburbs, such as Palmerston, Howard Springs, etc., can arrange collecting some of the many local woods that are very unique to the Tropics.
One of the good places to go for local woods, and not getting in troubles with the law or private owners by trespassing or cutting wood with permission (some tree are protected), is the local dump in Casuarina. This would require a chainsaw and someone with a bit knowledge about what's good and what's not.
A small charge would be payable at the gate depending on how much wood you would come out with but I remember that $30 would get me a 6"x4" trailer full...! This is obviously something that wouldn't be viable and or practical when traveling with a very short amount of time on land so, I would ask someone from that area to come forwards and help this bloke to get some samples of the wood available and local to the area...!
I knew exactly what to do, and I would be most keen in help if I was there, but I'm not...! I think that you may need to make some arrangements with some of us, Australian wood providers to get some of the Aussie woods you may be looking after. From me at the moment, you would only be getting what grows around here and in a 25km radius, there is at the edge of the Adelaide Hills (Fleureau Peninsula) in South Australia.
I hope that someone will be able to help you out...!
Good luck.
Cheers
George