I haven’t solved this problem yet either, and actual timed exams don’t figure highly on my syllabus either.
When I was an undergrad in the UK, many years ago, the deadlines were so strict that you lost marks for handing in late, and the number of marks lost went up by the minute/block of five minutes etc. So before the Internet, when essays had to be typed and handed in on paper, you would actually see the tallest, sportiest students racing across campus with other people’s essays as they could get there quicker.
Unfortunately, where I now study, the deadlines are pretty arbitrary. And I know that, so I can’t kid myself the date on the exam timetable is a hard deadline. No prizes for guessing what that does to my productivity …
I haven’t yet worked out how to put this into practice, but what I really need is probably an earlier deadline rather than an extension. So I would still end up begging for an extension but that would actually mean being on time with the official one.
It’s problematic though, because to make it work it would have to be real, but if would be a bit strange and probably not legal to have stricter rules for students needing accommodations. It would look bad in the press for the college too