Hi - I agree with what others have said:
1. If you're really not comfortable, you have the right to see anyone who will make you less uncomfortable.
2. Having been made very ill indeed by a famous female GP meno 'specialist' I have been very lucky to end up with a leading meno consultant who happens to be male. He is at least as comfortable/empathetic and a billion times more knowledgeable and has taken more & better care of me than any doctor (of any gender) I've ever seen. I'd go for expertise/approach over gender every time, but that's just me.
3. On reflection it is a sad truism (and indictment) of the system that there are probably fewer at the most-experienced consultants around, particularly perhaps in the NHS, who are female - but that's a philosophical/bigger conversation for another day! There probably is at least one junior woman in his team.
4. You have the right to ask to see a female doctor if you prefer. If you can get through on the phone, you're doing better than I've ever managed and personally I'd expect if you did you'd probably get some outsourced /offsite admin who a) wouldn't know and b) might move your appointment back further. If it were me, I'd go on the day and tell them when you arrived you want to see a female doctor. Clinics don't often tell you this but apparently you have the right to ask to see a consultant when you arrive - just expect them to make you wait and don't complain about the time - so I don't see it'd be any different.
When I went for a scan a few months ago, when I was finally called by a young guy, I said (as I had when arrived) 'No disrespect but I'd like to see' and he interrupted me to say 'a woman' - clearly because so many people did say that. I said no, a consultant (who in that particular clinic head consultant was a man and I named him). He almost fell off his stool because he is clearly so used to people asking for a female / female consultant, but I'd been told he was the best and that's what matters most to me. That was for an internal procedure, but I'm sure you're not the first to ask at this clinic. Just be prepared to wait, don't make a fuss about time - and I'd probably arrive a bit early and make clear you're happy to wait as long as it takes as they are flexing for you.
Good luck!
xx