I have no experience of going private, however, I had to wait from my gallbladder getting infected (few days after my scan confirming stones) in early November to towards the end of March before I could get through all the hoops of waiting lists, post op etc...
I had a call from the hospital about my operation date (which I had no idea of) to cancel it in the meantime too as they hadn't done other things they needed to do. End of December if would have been otherwise.
To be fair, I had to have an extra appointment with an anaesthetist (something to do with my arthritis) and they sent me to immunology in case I had a latex allergy (took weeks for that along!). I was pretty sure I hadn't, but now know about a number of allergies and condition I didn't know I had haha! Tested me for all sorts, which was useful, but did delay matters.
It was a very unpleasant time in between and by the time they took it out it was going septic and had split and I had gone yellow.
Obviously there were extra factors with me but if I could have afforded to speed things up, I definitely would have.