Oh my God. Never has a thread made me so pleased I live in Jersey where I can access a doctor (possibly even 'my' doctor) any day I want for the sake of a phone call. Never have to join a queue, never have to wait to have the phone answered, never have to wait more than a few hours (often even less than an hour) to get a face-to-face appointment, or a phone consult if that's more appropriate.
The catch? We pay to see the GP. (like I did for a UTI this week - phoned reception at 8am, was told Dr would phone at 9.40, which she did - spot on - didn't need to see me, prescription waiting 5 minutes later for me to pick up. And was free to fill).
It's £30 for a phone consult and £50 for face-to-face (but if you have private health insurance you can claim that back). Either half price, or sometimes free for children, depending on the practice, and people on local income support have reduced rates, or aren't charged at all.
All in all, I'm pretty sure that's cheaper than a private GP appointment in the UK, and better than idolising the 'free at the point of use' NHS. And it's only the GP we pay for when it comes to health care via the surgery. Referrals onwards to a consultant, and outpatients appointments are free (of course, you can pay to go private if you want, or have health insurance).
I don't get the resistance to paying to see the GP in the UK to streamline things with a small payment for those who can afford it, with a sliding scale of subsidies for those who can't. No, it's not perfect, but it's a hell of a lot better than the UK's 'lauded' NHS in many ways. The best of both worlds, really - GP, yes you pay, but after that it's all free.
The OP wouldn't have any of the issues she's having if she lived here. She'd have had a phone call with a GP within an hour or two of her initial phone call (which would have been answered almost immediately!).