This will probably be of no use, but I thought I'd share this in case it is.
My doctors surgery (used to be six doors down from me, now eight miles away) is impossible to get an appointment with. I need a fasting blood test, tried to phone and get an appointment, no luck. I've tried for about two months. They'd texted me to say I need this. Yesterday afternoon, I phoned again, listened through all of the options, selected a random one that was nothing to do with my enquiry and...drum roll here...got an appointment!
In short, if you select an option on the phone where you can speak to a person, then you're halfway there!