Have you had a copy of the admission authority's procedures for considering special arrangements?
Have they followed those procedures?
I'm doubtful whether admission authorities would normally make enquiries of medical specialists. They're more likely to take the view that it is for parents to obtain and submit their evidence.
They might perhaps contact the current school for information - but it depends what is set out in their procedures.
I get the impression that, generally speaking, they tend to be quite strict about allowing extra time (stricter than for other concessions).
Did any of the evidence you submitted specifically support the granting of extra time?