Hi Bombshell,
I can give you answers on what happened for us.
It was a pretty quick process from when we decided to pay privately to first round of IVF - I'm pretty sure we booked on for the following month. (We are in ChCh so might be different in other locations).
It will be up to the specialists as to what you need to do, I think they will try other options prior to IVF if its a possibility (wasn't for us).
Not sure about the payment thing though ..... we had to pay upfront when we booked on (or when we picked up first drugs), and that was it! We are onto second cycle and the same goes .... we're pretty broke now!
Hope this has been of some help! I'm sure others will have some info too