According to what I know you can only resit, Maths and English in November if you have previously completed a GCSE examination the previous May/June. All other examinations must be taken in May/June as there is only one examination period. If there was to be the option for GCSEs to be taken earlier, this would mean the exam boards would have to create several papers for the year for each subject. Past papers are only available for subjects outside English and Maths for the one examination period so this must mean I am right.
You could apply for a mitigating circumstance consideration to get extra credit, but whether this would be accepted by the exam board would depend upon how your illness affects your ability to work.