It depends on the set up in your local area.
I went to see GP at end of September. She did the referral letter, but said that the fertility clinic wouldn't send out an appointment until all the preliminary investigations had been done, they had the results and had reviewed them.
As the blood tests have to be done on certain days in your cycle, I didn't have those done until October and results back end of October. DH was referred to andrology for semen analysis and got the appointment through for mid-November. The results take 3 weeks.
Therefore it was first week of December by the time all the results had come through and were sent forward to fertility.
From this point it's been extremely quick. Where I live they run a first available appointment system, and don't let anybody book more than three months in advance.
Therefore, if there is a cancellation from somebody, they will reallocate that appointment to whoever phones up next.
So
1st week December - all results of preliminary investigations sent to the fertility clinic
21st December - letter sent out from FC saying consultant had reviewed results and I had to ring within 7 days to make appointment
5th Jan - 1st appointment with FC.
Like onoffon my first appointment was with the fertility nurse. She has referred me for an HSG, for which I have to phone on the first day of my next period to arrange. So that should be done end of Jan / beginning of Feb.