I did my first solo at EGTC (Cranfield) a few years ago, landed on 21 a tiny bit long but too late to exit at Charlie, so I announced I was exiting from Echo. I now know that I should have requested this rather than gone ahead and done it.
I may be wrong but is my understanding that you are simply assumed to vacate the runway via the nearest convenient and reachable exit. If you screw up the landing and can't make the first turnoff, you have every right to continue to the next. You don't need to inform ATC or ask their permission.
If ATC doesn't want you to take a specific exit, for whatever reason, it's up to them to inform you about it ASAP, preferably well before you turn final.
The only time you do need to ask permission is when you want to backtrack to an exit you've already passed. And of course it's bad form to slow down to a slow taxi speed when the nearest exit is still quite a distance away. Better let the aircraft roll at a high but managable and safe speed. (Taildraggers and such being the obvious exception due to their poorer ground handling characteristics.)