Some London hospitals are being quite tight on the reasons for allowing an ELCS at present due to politics being more important than patient care. Many hospitals are putting midwives and consultants under extreme pressure to reduce the number and unfortunately that is translating into patients being second best regardless of your medical history.
HOWEVER, having said that, if you have a good case for wanting/needing an ELCS on medical ground which it sounds like you do, then it does make it very difficult for them to refuse ultimately. It can just be a messy, unclear and slightly laboured process in getting to that point. They just want to make it appear that ELCS are not being dished out to a certain extent.
As it stands the NICE guidelines on CS are in your favour in that they support women who feel they want (never mind need) an ELCS. They are a useful thing to know and be able to back you up even though you have a very strong medical case here.
It may mean that you need to stand your ground and be forceful about it - changing consultants if you don't like the attitude of your current one is a good start. A good experienced consultant is far more likely to understand your concerns and needs (and to be blunt about it be more willing and able to justify to their superiors why you are having an ELCS).
I've seen a lot of threads of a similar nature, and it seems that there is an almost institutionalised trend to try and initially put women off an ELCS, often without proper regard for the circumstances. The good news is that being persistent about it and making your concerns clear does eventually led to you being taken seriously within the NHS.
So sticking with the NHS may not be as bad as you fear if you are happy to be assertive if necessary. You may not need to be though. As others have said, it could well just be a difficult/inexperienced consultant.
In terms of going private, I think the prices people have put on this thread are around the right mark, however have a good look into it as the prices quoted can be a little misleading and not include certain things which you don't initially realise (basically read the small print). I'd also try and make a decision quickly if you are considering going private as my understanding is that they can get booked up if you leave the decision too long as there is such a demand in London at present.
As for students. I'm at a teaching hospital elsewhere in the country. I have it on my notes NO STUDENTS. They asked at my booking in whether I was happy or not with them (in part because I suffer from anxiety) and were completely fine with this. They can only allow you to be a subject for students if you consent to it - just make your wishes clear.
Being honest, my attitude here really is that, if you have a need for an ELCS why should you pay for that? Why should the NHS make you feel like you have to go private? And under those circumstances, unless you would feel happier with the extra facilities that private might offer, and you are comfortable otherwise with your NHS care to stick with it. You are within your Rights to expect the most appropriate care for your circumstances under the NHS and if you have a consultant that you feel isn't looking at your medical history properly, you are in no way being demanding or difficult in asking for a second opinion. Its a question of competence and responsibly as much as anything, and unless they are held accountable for poor care then it doesn't improve for you or anyone else.