Where have you got to now, Mrs N? What sort of doctors have you seen and what sort of referrals have been done?
It would be normal, I believe, for the GP to refer someone who suddenly has part of their visual field drop out to the nearest Opthalmology Outpatients as an emergency case. Leastways, that’s what happened when it happened to me. So that brain MRI was commissioned by Opthalmology, although I already had an MS dx, so the MRI was just to confirm everyone’s assumption that it was MS and not something else that was causing the visual field defects. An emergency Opthalmology referral would tend to get you seen much more quickly than a non-emergency Neurology referral, for instance (that can take months). If you have not been referred yet to an emergency Opthalmology Outpatients clinic at your nearest local hospital that does one, I would be asking your GP why not. If he/she feels confident that there is nothing too much to worry about, then you can take some comfort from that of course, but it is perfectly reasonable for you to ask the question.
Generally speaking, speed is not of the essence when it comes to an MS dx. The urgency (when there is urgency) tends to come from the need to exclude alternative explanations that (unlike MS) are emergencies that would require urgent action - some eye things fall into this category, as do some brain things and some other things besides. If your GP thinks that you ought to see a neurologist, but does not think that there’s any rush, then going private is an option, as you say, if you want to save some waiting time. If you select the ‘Near Me’ orange tab at the top of the page, it will give you information about the neurologists in your area who practise privately and who are also NHS neurologists. So you could ask your GP to refer you to one of these people privately. What tends to happen is that anyone who looks (to the neurologist) as though they might have MS (or some other neurological condition) would then get transferred to the consultant neurologist’s NHS list at some point, either before or after investigations like MRI, lumbar puncture etc. That’s what happened to me (I had private health cover at work at the time, but would probably have put my hand in my pocket if I hadn’t).
I hope that you get some answers soon. Please do not be too quick to jump to the conclusion that you have MS. There are lots of things that cause MS-like symptoms. Most people who worry about having MS turn out not to have it. Of course, I haven’t the slightest idea whether you do or whether you don’t - I’m just saying that you do not yet (from what you have shared here) have grounds to assume that you have.