Welcome to the forum. Obviously we’re (mostly) not medical professionals so can only offer opinions based on our own experience.
Notwithstanding that, I should imagine that if you are ultimately diagnosed with MS, it would be the relapsing remitting type. The reasons I’ve come to that opinion is that firstly that something like 85% of people with MS at least start out with RRMS. Also, your initial symptom responded to steroid treatment and you had at least partial remission. That’s a good sign.
I would think that if you do have to wait for a neurology appointment (and accepting the diagnosis of MS), ultimately you would get at least partial remission from intervening relapses. But early diagnosis and subsequent disease modifying drug treatment would be expected to reduce the number and severity of relapses. So quicker diagnosis is definitely preferable.
But, having said that, going private can be a very expensive start to diagnosis. It’s not the cost of an appointment with a neurologist that’s going to break the bank. It’s the cost of tests. As you’ve had an initial MRI that might bring the cost down, but it may not have been sufficient for a complete neurological diagnosis. The neuro might want a spinal as well as brain MRI. They might also want MRI with contrast dye (this will give much more information about inflammatory activity). So you might still run into costs for MRIs. Also, a lumbar puncture is often needed which has a further large cost associated with it.
Having said that, further tests could be done on the NHS. If you happen to find a local neurologist with both a private and an NHS practice (preferably an MS specialist), it’s possible that you could see them privately and then get easily slotted into their NHS practice. They may not be able to circumvent the existing waiting list times though, so it could be a rather frustrating way of spending the waiting time.
I suspect it really comes down to how much you can afford to spend. Perhaps before you even consider this, you could have a conversation with your GP. Ask whether in their estimation you are likely to be diagnosed with MS. Also, whether they can re-refer you but class the referral as ‘urgent’.
Failing this, you could just try going to A&E. It may get you no further. But it might. Given how concerned you are about your eyesight, it may be that you could see a neurologist who thinks a further course of Prednisolone is warranted. Certainly steroids are considered very useful for relapses involving eyesight.
You should be aware though that steroids work best when they are taken right at the beginning of a relapse. Taken later you may find that you just have the horrible side effects from the steroids but no quick recovery from the symptoms.
Best of luck with whatever you decide to do. I hope you do manage to see a neurologist as fast as possible.