Hi
I'm in Adelaide. Been here since december 2011. I came over as a gardener/tree surgeon and got 2 job offers (after applying) within days of starting looking.
I went to a company called MVS here in Adelaide, great firm good lads and plenty of work. They took me on as a casual no problems. I did decide that it wasnt the career for me anymore and have since secured something else. However another guy who got taken on at the same time as me was over on a working Holliday visa and was hoping to get sponsored by the firm, according to the lads who worked there the boss was more than willing to do this so long as you were a good worker.
There are loads of tree companies (tree loppers) out here not just in Adelaide and the Aussies love their trees so there is no shortage of work. Most will have come into contact with poms and will know what the NPTC certificates are and will have heard of colleges in the uk such as Myerscough, they did at the ones I spoke to anyway. But they are more interested in whether you can do the job at the end of the day and if you have your own kit (climbing) all the better.
Check out http://www.arbjobs.com usually a few on there.
Hope this helps. Good luck.