It really depends on your situation, the area you live in, etc. The best thing to do really is to contact the clinic and ask them what their waiting times are like.
As an example, this is our journey...
Jan 2021- realised we were having problems conceiving, contacted fertility clinic directly for tests. We went private as the NHS were taking 3-4 months to do tests.
March - All tests had been done and we had our first consultation.
They recommended we would need ICSI but also sent us for further tests. They also referred us to the NHS consultant as we were eligible for ICSI on the NHS. There was a 3 month wait for the NHS consultant appointment.
July - Initial call with NHS consultant who has agreed we need ICSI and will now request the funding
Funding takes 6-8 weeks to be approved, so the earliest we will start is September. That's 9 months from initial contact but bear in mind we paid for private tests.
If we hadn't paid for private tests I think it would have been around a year from initial contact to starting NHS treatment.
If you have private treatment as well you can start quicker as you don't need to wait for the funding approval.
This will all probably vary hugely depending on your area though. Really you will just have to contact them. Good luck.