I have not heard of this being the case in Birmingham. I believe that could be the case in Warks but not Bham.
There will be a late sitters exam usually the Wednesday after the test; this is for children who, for religious reasons (not taking exams on Saturdays) or through illness, could not attend. In each case there has to be documentation from a religious minister, or if ill, from a doctor, stating the issue.
If a child has a longer term illness and needs to wait longer to take the exam, again this has to be backed up with medical evidence.
The proportion of children taking the exam late is usually tiny.
I absolutely echo the sentiment about not posting information on the forum and being very wary of responding to PMs asking for information.
However posters are welcome to send information to KenR as soon as they wish; he will keep this confidential until January time and will then publish it for the benefit of parents next year.
As for not disclosing information to tutors, I would hope that the vast majority of them would abide by a clear code of ethics and would never pass on that information to a late sitter so I don't fully support that point. It seems reasonable for them, if they have built a relationship with your child, to understand how they found the exam and to take feedback on where their provision could have been improved.
Fair enough about relaying feedback to your own tutor but again I wouldn't recommend it until next January to be on the safe side. Again write down all info just after test then pass it on to your tutor a few months later.
Majority of tutors are scrupulous but it's best to be on the safe side.