I always thought the exam papers 'belonged' to the examiner, not the school, CAA or student. The examiner is sort of independant of the school - as in, they have to have their own license and renawls etc. issued by the CAA to be able to conduct exams.
There is a single sheet of paper which goes to the CAA on completion of the course which has your hours summarised, and a list of exam marks - each line signed by the examiner who marked the particular paper. I reckon you should get your current examiner(s) to sign off each exam on this form now - even if you go to a new school it's the original examiner that'll have to sign this form for the exams you've already done - not the new one.
I don't think your student records go to the CAA - just this form, which sumamrises things such as total time, solo and dual time, cross country time etc. along with exam marks.
Hope this makes sense, and hope I've got it right.