Nazly or anyone else, if you have very severe anxiety, to the point that its affecting your day to day life (eg: dominating your thoughts or giving you nightmares for example), please talk to someone about it.
I can't guarantee the response you get if you come right out and suggest an ELCS as it is very dependent on individual hospitals and individual staff you come across, however you may be able to access perinatal mental health support at the very least.
I was very dubious about it, but its helped me in the sense that its alerted them to the fact that I need additional support. I've personally not gone down counselling route for various reasons, but it might be appropriate for you. Making the perinatal mental health team aware of my problems has at least given me the opportunity to discuss aspects of care that I don't think I would have otherwise. Depending on how open minded your hospital is, you might be able to discuss alternatives and potential scenarios/plans which take into account your anxieties.
If you did feel that an ELCS was right for you - and I really don't think its right for everyone with anxiety - then going down the mental health route is generally the way to do it anyway. To give you your best chance at getting one, right about now would be the time to get the ball rolling.
The NICE guidelines on maternal request CS are particularly aimed at women suffering from anxiety. It IS considered a medical reason to have an ELCS in the UK. My consultant midwife has made a point of stressing that he regards my preference for an ELCS is a medical NEED.
However, I do know that a lot of women are struggling to get their case taken seriously though, and its not uncommon to come across a lot of hostility to it from HCPs. I don't want to put you off, but I think how hard it can be should be mentioned. Ultimately though if you are pushy and persistent, can show you understand the risks and are willing to put up a fight then eventually it is usually possible to get one agreed. I've only seen a couple of women on mumsnet refused and had their appeal also refused; the vast majority do get the outcome they want. The big thing, is it can be stressful and uncertain process in its own right and you may need to be quite committed and certain in your choice.
There are quite a few threads about it in the childbirth section, if you want to read up a bit more or ask questions to a few more people. There are a few people on their with experience of the system which is different to mine. The full NICE guidance is available here. The relevant bit is from page 96 with the key recommendations found on page 102. (Be warned there is some flaws and ommisions to what NICE say which are important and should be considered - most notably their data on risk is restricted to women having their first child and there is no comment on risks for subsequent births)
I don't like to 'advise' anyone one way or another too much as its such a personal thing and I don't think there is a 'right answer'. But feel free to ask questions, or ping me a PM if you want.