I'm currently pregnant from IVF cycle done at St Mary's Manchester 
From GP referral to first appointment it was around 12-14 weeks I think, but this was longer than usual we were told as I believe they were exceptionally busy at that time, so fingers crossed it is much faster for you!
However, when we received our first appointment letter - it included details about booking in for fertility tests. (This was 2017 so may have changed since. But if not...) This included bloods for me & a scan. Sperm analysis for DH (even though we had one done through the GP)
The first appointment occurred after these tests as they like to save time and go through the results with you and plan how to move forward.
For us it was male factor issues so we were offered IVF. Everything moved very quickly after that appointment and there were no waiting periods at all. We decided to postpone our follow-up/IVF plan appointment until Sept 2017 so I could lose some weight.
Once we had the appointment we got the go ahead to ring up on next AF to request IVF treatment. They can refuse you 2 times if they are busy.
I rang up late Sept 2017 and they used that month to deliver medication and for us to have a 'teach' appointment for the injections.
I began injecting for short protocol in late Oct 2017 (when the next af arrived).
Late Nov 2017 we had the embryo transfer.
Early Dec 2017 I had a positive pregnancy test.
Sorry if I went on a bit! Just wanted to give you an idea of timelines.
The staff were lovely, so very welcoming and friendly. We are actually having our ante natal care at the other part of St Mary's across the campus because of how well we were treated.
If you have any questions at all, feel free to PM me!