We have just been through this battle.
DS had extra time in exams and was allowed to type on laptop rather than writing by hand. Although he was assessed a couple of times before the age of 12 for dyspraxia or similar coordination difficulties, he did not have a statement, or a diagnosis. School just put in place the alternative arrangements for him.
When he started Uni we - like you - assumed that because he had had these arrangements at school, that it wouldn't be an issue. Except it was very much an issue. The University disability and wellbeing service insisted on a formal diagnosis before they would even start the conversation about additional arrangements or using laptops or extra time. And they would not do the assessment. So it was a mad scramble to find the appropriate professional, and it cost us £750 for them to tell us what we already knew - that he has significant issues with fine motor control, and that those issues coupled with poor organisation and time management strongly suggest dyspraxia. (Or development coordination disorder, which is the new name for the same thing).
Once we had the piece of paper then doors opened. He has his additional time and works on a laptop, he has a named disability advisor to go to with problems, and is offered software and courses to help him with organisation, planning, revision.
So the key take away - do not assume anything. Ask the explicit questions of the university and if you are in the position of needing to get a formal diagnosis, start the process asap. Good luck!!