Funded or private?

After 12 months trying to conceive and a few cycles where I haven’t ovulated, our doctor is referring us to a fertility specialist.
We have no idea what to expect or how long the process takes. We are in Christchurch and wondering whether we should consider paying for our first appointment. Does anyone know how long the waiting list is? Do fertility Associates and Genea Oxford both do public and private?
My husband has done sperm analysis and come back clear. I have already had an ultrasound and progesterone bloods and various other bloods over the last year. We are aged 32 and 33.
Has anyone been in a similar situation? What advice do you have?
Thanks in advance.