Minus the "illness and few other things" would you have honestly and realistically gotten higher grades? The entry requirements are there for a reason- simply, if you can't achieve them at GCSEs, your chances of passing that subject at A-level are grim. However I know a girl at my collage who also didn't meet the requirements at GCSE due to illness but as she was predicted higher grades before, the college allowed her to study her chosen subjects anyway and she retook the needed GCSEs in November. Another only sat her English and Maths this summer, I'm not 100% sure why but similar circumstances I think (she is studying maths A-level so found the GCSE a breeze).
This depends on your chosen sixth form, I appreciate the one I go to is incredibly supportive when it comes to odd study programmes, retaking GCSEs, mixing second and first year courses etc. but maybe talk to them about it.