Any idea that one might have at the start that quality of thought and quality of handwriting broadly go together soon has to be abandoned. Although a controlled and disciplined hand rarely appears in a very weak or brief response (unless, perhaps, the candidate has been taken ill) - at the other end of the scale, poorly-formed and uncooperative handwriting is quite often the vehicle for answers of merit, detail and sophistication.
The word 'illegible' doesn't precisely describe the untidy, challenging, rough-and-ready scripts that I was poring over - they were not illegible, as the phrases and sentences, could with effort, nearly always be fully worked out - everyone, after all, was writing more or less the same things about the same poems. But my overall impression from reading page after page was of an underdevelopment of the handwriting art for many of these writers, lack of skill and consistency in forming writing, the long-term effect of insufficient practice or emphasis in the system and society in which they have grown up.
The keyboard or keypad makes its entrance early in students' lives, and handwriting, which may still be introduced to learners as an art or craft in the first years of school, perhaps does not retain that impression of being a valuable craft for as long as it did in earlier generations. After all, out there in the world, handwriting is not so necessary any more - and indeed is even discouraged as an old-fashioned form of production, inconvenient for reading and sharing. Who wants it? Who writes assignments, letters, notes, memos, by hand any more?
Except that, for the time being, we still have exams written by hand, and we set great importance by the results of such. Most schools, I imagine, crowd their Spring revision time-tables with 'timed essay' exercises, and doubtless these have a beneficial effect on students' readiness to answer fluently in the examination setting. But not, I suspect, an equally beneficial effect on their handwriting: possibly the reverse. There are of course no marks deliberately awarded or deducted for it - examiners are required to guard against penalizing the kind of handwriting that merely makes their task of reading harder and slower.
Students all through school are endlessly reminded that neatness is important, and lack of it will cost marks. In Maths this can quite readily be proved - messy Maths working (as I know from tutoring) is rarely error-free. In the humanities, the importance of neatness is more of a supposition: willy-nilly, poor handwriting must have a negative psychological influence on an examiner, irrespective of the quality of the thought, just as a partisan home crowd must negatively dispose a referee against a visiting team.