I opened this thread as my DD is y11 and I am thinking of trying to arrange a formal assessment as she has a collection of varied difficulties.
However, I suspect the previous posters are right, there is not much chance of anything being put in place for GCSEs. The best you can probably hope for is for access arrangements for A levels.
DD has slow processing. This was assessed in y7 and re-confirmed in y9. She qualified for extra time, but then lost it for some subjects as she wasn't making good enough use for it. There are tests the SENCO can run, they have to score below a certain level to qualify for extra time, and then it has to become their usual way of working, with benefit shown (eg moving to a different coloured pen).
You are right that his high cognitive ability will have helped mask slower processing but as the work gets harder it causes more issues. I honestly have no idea whether bottom 13% for processing would be 'slow enough' to qualify for extra time.
I would
a) contact SENCO / HoY to inform them of results (and why you organised the test) and enquire whether it is too late for them to assess for extra time for GCSEs
b) if he is staying on for A levels, ask them to assess anyway for A levels
Unfortunately, if this is the first they have heard from you my guess is you won't get very far for the GCSEs, but it may still be beneficial for the A levels.
I am myself in the process of reviewing the info I have had on DD over the years to then ask the school & college SENCOs about further assessments for my DD as there is still more going on than we understand.
Hindsight is a marvellous thing.