This is a very difficult situation to be in and I'm sure everyone reading it will empathise with you.
As far as I can see from looking at the government websites the Local Education Authority has a statutory duty to provide a minimum level of education.
If you are unable to attend school through certified medical conditions, then the Local education Authority is obliged to make arrangements for your continued education. This does not have to be at your current school and it can be private home tuition for instance. So the school is within it's rights to remove you from the admissions register so long as they have agreed this with the LEA and the LEA has made proviion for continued education.
However, it seems because you are no longer of compulsory education age any provisions the school makes for you is entirely at their discretion and goodwill. It seems that they have already done this as you are effectively in yr 14 now because this year is a resit?
To have a case for continued education, you would need to prove that the school and the LEA failed to meet the statutory minimum education requirements. Doing this will be a long and protracted process so you have to make a decision: is it in your best interests to go down this path with no guarantee of success and waste a lot of time which could be better spent on your actual education albeit under your own expenses.
I'm not qualified to say whether you have a case or not. But you should definitely seek independent advice from the LEA as you have paid for your resits and they will be obliged to allow you to sit them under controlled conditions. That ironically may be within your current school, but you would be an independent student and therefore will not appear on the schools statistics if that makes sense?
Definitely contact your LEA though.
This document (whole and especiall para 18 and 19):
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/269469/health_needs_guidance__-_revised_may_2013_final.pdfSection 8 of ths document:
http://webarchive.nationalarchives.gov.uk/20130401151715/https://www.education.gov.uk/publications/eOrderingDownload/UKSI-20061751.pdfHope this is of help.