Exam extra time

I've been told by the support team at my school that I cannot get extra time for my exams because I don't have a cognitive impairment.
My problem is that I'm on a really high amount of medicine (the specialist I saw recently was actually shocked that I was put on them but that's a whole different thing) that severely impact my processing speed and just overall make me slower. I have also developed a severe tremor in my hands which also slow down my writing speed.

I have been told I can have rest breaks which obviously cannot compinsate for the tremor and a laptop for exams which I don't think will help much either.
I've been told by multiple teachers that they think I'd be able to get extra time but the woman in charge of it refuses.
I'm wondering if anyone has had experience with this or a similar situation as I'm really at a loss and not sure how I'll cope.
Thank you!

How did the support team at your school reach the conclusion that they did? Were you asked to complete a series of tasks, like remembering a sequence of numbers, or writing down answers to some simple questions?
If so, were you medicated at the time you took the assessments?
If the answer to both questions is yes, then either the medication doesn't impact your processing speed in the way that you think it does, or it doesn't impact you sufficiently to meet the threshold for extra time being awarded.

They haven't carried out any tests but they knew me before I got ill and say I'm 'too smart'. They did mention testing but they said it's not going to get me anything. I'm not sure if they'll do anything about my tremors becauses it's not cognitive but it makes it so hard to write (almost like a parkinsons tremor when it's bad). I guess I just have to make it work.

There are many "smart" students with below average processing speed, and if it falls below the threshold for getting extra time, then they'd still be eligible. (I've said "below average" here, but for extra time a student must have "cognitive processing/fluency" which is quite a bit below average - the the rules themselves are quite complicated.)
With regards to your tremors, using a laptop might actually help (although I know you're not convinced). Another theoretical option is a scribe (where someone else does the writing for you, based upon what you tell them to write). I don't what what the threshold for this is, but I suggest you try using a laptop in class (or at home) first and see how you get on.
Your school's SENCo will be at expert at all the rules and arrangement available to you, although I don't understand how they've managed to establish that you "don't have a cognitive impairment" unless they've run the specific tests for that.