OP - have you considered ringing 111 back and updating them? If you are alone in the house, with no friends/family able to get you to A&E and are fainting when you stand up they normally would send an ambulance - I know because the same thing happened to me a week ago.
I had a number of other symptoms including struggling to breathe - it was, as I suspected, a flare up of a chronic illness I have and I ended up staying in hospital for a week, and got all the support I needed until I was safe to come home.
The system is under huge pressure but it isn't broken, if you really need care you will get it, it just might not be as smooth or as painless as you'd like. I waited 3+ hours for an ambulance (which I thought was pretty good, although I do live near two major London hospitals which probably helped) but once I got there, even by ambulance, there was no bed and I wasn't assessed by a doctor for 4 hours.
I get feeling like you can't wait in A&E but you may just have to, even if it's in a separate bit for people bought in by ambulance. I was in agony sitting down but told it wasn't safe to stand and I couldn't lie on the floor, I sympathise with you deeply, I cried when I finally saw a doctor and got an injection of morphine for the pain (unable to keep any fluids down at all). I'm not trying to scare you off in anyway, if you need to be there you need to be there, I just personally find it helpful to mentally prepare.
I think if you really feel so unwell you can't get to A&E yourself and you don't want to wait for a call back you'll need to call 111 again and try to explain further, and you're best to get on with it as the process will take hours whichever way it happens unfortunately.
Benefit of A&E as PP said is if you deteriorate quickly, you're already there. I know you're concerned re a taxi, although to be honest I would have tried to have take one had I not been vomiting continuously and have a history of seizures as well as just fainting, so that may be a factor for you, but do you have a nice local firm where you can speak someone and explain? Could the driver park up and assist you down any steps and into the taxi and walk you into A&E the other side? Presumable only a few steps each way with someone to lean on? You may find it quicker.
Hope you feel better whatever you choose to do.