GCSE Maths Early

Hi, I'm currently in year 10 and excelling at maths in my top set + higher maths class, moving on to calculus in my own studies and in competitions. I brought up my individual study with my maths teachers, and explained to them that I was struggling with self-study because of home situations and other school subjects. I proposed the idea of doing my maths GCSE early because I frequently get near if not full marks in exams and past papers and have covered all of the subjects, to which they said that it was a possibility and they would ask the head of Maths, who also said that it would be possible. I also went to my careers leader for advice since maths, specifically the pure maths taught post-GCSE, is something I would like to go into for my career. They said they would ask the deputy headteacher about it too. So it seemed like the school would allow it, especially because my friend's older sibling did the same thing? The head of maths recently spoke to me about the school saying that it wasn't allowed because of certain policies - it didn't seem that way when nearly the entire maths faculty at my school said it was possible and when they let someone else do it, so I was really confused. I asked my careers leader who told me that this may be because the entries for GCSEs for year 11s had probably already all been completed, and it was too late.
I feel really disappointed and hopeless for my success as a mathematician - nearly all of the best mathematicians I've seen completed their stuff early and got straight to work as younger children and teens on mathematical research, and I feel so behind. I don't think anyone here can offer advice but it's worth asking.

Tbh, Id not worry that much about taking it early or not. It means little. The main thing is to do some "harder" stuff yourself and put some effort into that. So learning more elementary stuff/problem solving as well as finding out about stuff like calculus. For the latter, you can look at the history (so archimedes onwards), its influence on the development of functions, continuity, ...

I'm only worried about it because I see so many other successful mathematicians doing their school exams early and I feel so behind compared to them. I really want to be like them but I'm just not good enough and everything is in the way of my self-study. If i did my GCSE early, at my school I would be in Further Maths for the duration of my other GCSEs (a year in my case) and I would have more time to focus on supercurricular maths. You have a good point but i feel so inferior and I don't think i'll make it in the world of mathematics otherwise.

Some do it early, some dont. Id guess its more the latter but youll get 0 brownie points either way in an oxbridge application (for instance) as they simply dont care. Similarly for a further maths gcse. You could do both normal maths and further at the same time in y11, you could learn (parts of) further maths and not do the exam or ... If youre capable of getting 9 in y10, then it wont take much to keep your skills sharp and do it in y11 and do something like further if you want. If your school requires you to do normal gcse maths in y10 before doing further in y11, then Im not sure why theyre being reluctant? But it not necessarily a big deal in the long run
As in the previous post, Id not be bothered and if you want (your school does/supports) to do further maths in y11, just go ahead. If your school doesnt do it, why not do some stuff yourself that you find interesting. The further maths stuff will be covered in maths/further maths a level and you have enough time to learn it then, so there is no need to do it early. So depending on what youve done already, maybe find out a bit more about elementary number theory, history of maths (as its a good way to learn it), have a sniff around calculus (so precursors, ...), a basic idea of functions, integer/diophantine equations, mod arithmetic, ... You could as well see if there are any maths clubs youre interested in, so parallel, ...Youve said a couple of times that self study is difficult, why is that?

Self study is difficult because of my situation at home, alongside loads of homework, projects and tests at school for other subjects. I've been trying to get a grasp on calculus since year 8, and I feel like if I could have managed I'd be a lot further. This is one of the only reasons I want to do my maths GCSE early, so that I can learn the things I'm trying to self-study in a classroom like all my other subjects.

Id have a chat to your maths teacher / deputy head about these things, especially if there are issues at home as you seem to be alluding to. If there are problems about working at home, it would be better to try to sort these, rather than getting stressed about doing gcses early.
As before, there is no need to complete maths gcse early to apply to oxbridge++ and neither is there any need to do calculus early, so if you dont get this sorted now, dont worry.
A few general comments. For oxford, mat (entrance exam) hasnt concentrated a lot on calculus and while the test changed a bit this year, you can see the past years online. So there really is no need to learn calculus early. Doing a bit of elementary maths/problem solving (ukmt, ...) is useful to do harder questions and a reasonable number of the mat questions have an element of it. Similarly for having read of some pop maths, history, ... books to get interested. "calculus made easy" is a classic book and freely available, and while its a bit wordy in places, it concentrates on understanding in the first few chapters. If youve been trying to learn it for 1-2 years, then maybe give yourself a break? There are plenty of problem solving/elementary maths resources.
Id really try and get some study time sorted (home? or school) but I cant see that pushing yourself to do gcses early is the way to go.