You should be able to arrange your situation so that you can graduate before going to the Academy. They give you 90 days to submit all elements of the CIL, which includes things like background check, drug test, medical exam, and so on. Most people complete it much faster for obvious reasons, but if you just slow-walk the process I think you'd be very unlikely to get a date before May. Assuming that you don't get the CIL until late January, and you take most of the 3 months to complete all the requirements, then the absolute earliest they'd be able to call you with a class date is late April, and class dates tend to be issued at least 4-6 weeks in advance. And that's all assuming the best case - in all likelihood, medical will be backed up and will take at least a month to review your file anyway, which would make all this moot. And of course, you could receive the dreaded Tier II, in which case you'd be lucky to get to the academy before NEXT May.
So yeah, you can manage the situation so that you should be alright. The hardest part will be resisting the urge to get every requirement completed immediately, but if you slow-walk the process you should be fine.