I think Big Grecian did actually answer your question.
According to your first post, the Competent Authority that holds your records is not satisfied that you completed the integrated course in time and say that in addition you must resit the ATPL exams. On that basis right now no licence will be issued. If your exam passes are not valid ..... no licence.
As far as rules go, though, I think the UK CAA are correct. If you look at the reference in Part FCL that Big Grecian gave you you will see that exams are quite clearly valid for 36 months from the date of the last exam pass. I can see how the Competent Authority might insist that the integrated course be completed in three years, I can not see (i) why they do not extend the period as they are allowed to do (although they clearly don't have to) or (ii) how making you retake the ATPL exams which you have already passed and appear to be still valid is in any way going to rectify you going over the three year period for an integrated course.
Big Grecian is correct, you need to be talking to the Authority that had oversight of your training, I would politely ask them the questions in (i) and (ii) above.
PS I am not sure whether the Authority are actually suggesting your ATPL exams are invalid, I don't think so. If they are, if the exams were not valid in March 2015 you should not have been able to take the CPL IR skills tests.