Hi hanshan, we had all our investigations undertaken at Hammersmith/Queen Charlotte's. I'm not sure what London borough you are based in and if you are NHS but we were Hounslow and this is the only option we were provided with.
Our experience was once they conclude that you need IVF/ICSI and are in a position to start the treatment there was no waiting list. Unfortunately by the time we got to this point we were just about to move.
We first saw the consultant 3 months after the GP concluded we needed to be referred. However even getting to the point of referral took a few months due to waiting times for husband's tests. We needed quite a lot of further tests along the way adding to the time so not sure if our experience is typical but we knew from the first appointment with the consultant that ICSI would be required.
To be honest were quite surprised by how long the whole process and some of the waiting times between the tests. Saying that when I enquired about private appointments at IVF Hammersmith you could pretty much have the pick of appointments with any consultant within a week or two. Good luck and let me know if you have any questions.