I tend to find the easiest way to get a doctors appointment, if you can't get pass the receptionist and rather than going to A&E. Is to complete an online form with your doctors practise. They then have 48 hours max to reply to that form, I describe what they do as triaging it - so the receptionist receives the form (I assume), then a decision is made in which direction does it go, doctor, nurse or is it just simple administration.
Generally you will know when you send the form if it's the latter, you know what's wrong with you, you have history, you know what you require in way of help, it may even be much simpler matters than that.
You may even know the doctor actually can't help you, because you are under the consultant, then it's their secretary you need to be contacting or that department within the hospital, not the doctors practise, but a lot of patients may not know that.
Other than that, a nurse from the doctors practise may call you, or your doctor may even call you back within an hour or two and ask you to come in that same day, hence you've bypassed the receptionist. You just sit at home and wait, which is far better for you than sitting in A&E, you will also have had confirmation your form has been received.
Where it can fall apart is if the hospital department you are under is under extreme pressure, then it just becomes a knock affect to everything within the NHS ie the snowball starts rolling, which is most times and people end up rolling into A&E. Some patients don't even know what's the best route to take, maybe they've not been ill before, this in itself can cause problems and frustrations to the patient and in itself generates complaints.
So as Labour did last time, you need to get the wait times down in every area, but they should piss the next election, easy peasy, so the future looks bright.