Anna, my son, also attempting exams in October went through a similar phase of loosing confidence as soon as faced with timed tests. Although I did not have the benefit of a tutor, two things helped in our case:
(1) Do not allow the clock to limit the amount of paper he covers, you need to let him complete the paper and the speed will come with practice - the key thing here is that he gets familiar with the type of questions and the format/ layout of the questions - quite honestly practice, practice, practice is the the golden key
(2) in my son's case I had the good fortune of finding boys who had completed their exams this time last year - it was fantastic - the 'graduate boys' found their own ways of explaining 'tricks' to certain questions and soon the companionship was getting results, and very encouraging for my son - so much so that in the end he was keen to take the other kids on in a timed test!
Hope my experience helps.