I had a planned cs on the nhs for my 1st dc for fear of childbirth, and will be having a second planned cs shortly.
I had to have several meetings with MWs, consultants and a mental health specialist before it was agreed. They were all helpful and sympathetic btw - but (as with much nhs care) this seems to vary from place to place, and there have been posts here from women who have been treated a lot worse and had a fight on their hands.
I think it is best to assume that you cannot have one just for asking - and that if you are genuinely very fearful, they will want to be convinced you are scared to the point of phobia before taking it seriously and going down the cs route. I would also make it a priority to go 'beyond' mw care asap. MWs cannot in any case agree a cs for you, most are very anti-cs and it's not their job to offer a psychiatric evaluation.
btw, it is always worth doing your research about risks etc as far as you can with cs. Asking on boards like this is fairly pointless - you find women who have had horrific vbs and women who have had awful cs's. Emergency CS's represent the majority of cs's carried out in the UK, and this tends to be reflected in discussion. My planned cs was calm, relaxed, well managed and as long as you keep on pain meds afterwards, pain need not be an issue. I was home after 2 nights in hospital (have been told I can come out after one night this time if all is well) - recovered very well, bf-ed my ds successfully and after I had my stitches out 5 days post op, felt really good. Friends having 1st babies vaginally around the same time had a variety of experiences - but nearly all of them needed a longer stay in hospital than I did and had ongoing problems afterwards (epsiotomy infection, restitches, readmission to hospital). A couple are still having physio/counselling 2 years on, which I find a bit shocking.
But that's just my experience. I think you are best off asking specifically for advice from women who have asked for or had cs for fear of childbirth (some will have had cs, some may have had counselling/vb). Otherwise all you'll get is general responses about cs, and these threads tend to fill up with people who have not had a phobia about birth, but had emcs's when they desperately wanted vbs. Unsurprisingly, they are very anti cs! and tend to assume their experience is the 'one experience' of cs, iyswim.