I went through this when I changed schools last year...
This is my understanding:
'Student records' belong to the school.
Exam papers/results belong to the CAA (not you or the school) - this includes things like the QXC report.
If you change school, the exams etc. will go with you, the other stuff may or may not - but it doesn't really matter.
In practise, I think even the dodgiest schools DO tend to pass all the stuff on (as, by definition, they don't want the CAA asking questions).
If you feel you have reason to change schools - just do it ! I made the mistake of hanging on with a very dubious outfit and it cost me a lot of money and uneccesary stress.
p.s. If you can get your instructor to 'sign off' things like solo hours in your log book before you leave, it may make life simpler at the new school.