It was a while ago now, but I was referred to UCH's Reproductive Medicine Unit. If I remember correctly, DH and I were seen within a couple of months (sorry, can't remember exactly, but I'm sure it was no longer than that).
I don't know which number you've been calling, but you could try these: 020 3447 9759 and 020 7380 9697, or email [email protected] or [email protected] - if you get no joy there, maybe try the main switchboard and ask to be put through. I do remember it was difficult to get through on the phones sometimes, so it seems as though things haven't changed much.
I'm really sorry to hear about your losses. My situation was different from yours, and so the procedure might be different, but my initial tests were carried out fairly swiftly, as they were just blood tests. We were seen by Mr Salim, who was really lovely and made us feel very reassured. He talked through everything with DH and I, took lots of info and outlined the tests that we would be taking and what they were for. He also told us about the treatments available at UCH.
We then had to go to the blood place in the main building (I can't remember when DH gave his sample), and I had to return after I had my next period (can't remember exactly how many days after) for further blood tests.
I had to wait quite a long time until my next period arrived, which was very frustrating as I was anxious to get all the results. Once it arrived, I had the additional tests and saw Mr Salim soon after for the results. They were clear, so he arranged for further tests which meant coming in for day surgery. Again, this appointment didn't take too long to come through but I was told I couldn't confirm the appointment until I had a further period, to rule out pregnancy.
That period never came, and I (finally) got my BFP. I ended up having an ultrasound on the day that was provisionally booked for my op.
I think they do HyCoSy at UCH, but I'm not 100% sure - sorry.
Best of luck, and sorry I can't remember the timescales more clearly. I just remember that once the ball started rolling everything happened quite quickly and smoothly. I hope they can see you soon.