I sat my GCSEs last year and what basically happens is you come into school, collect your exam results and if your school has a sixth form attached to it then you open your results in school so that you can be there for registering for the sixth form also.
I think you can be reconsidered. From what I know, on the day of results, it’s also the day of registering so you have to rush to the sixth form you want to go to.