It is entirely possible, although not advisable. I had a student come to me (from a major school that shall remain nameless, he was not one of ours) desperate to pass his final ATPL exam, having passed his CPL and coming up to the 6-month limit for the pass to licence issue. This is unusual : wait for the results before your CPL skills test, even I would suggest befire getting too far with your training unless you are supremely confident.
Then wait and apply for the CPL and IR at the same time - I did, and I seem to remember it is a little cheaper this way (they only print and stamp a piece of paper once, which only costs £180 or so
)
Are your school not flexible about your IR start date? You can be delayed by any number of factors (I was ill for 5 weeks in my CPL). If you miss your start date and rebooking is awkward try other schools - my employers would certainly try to help!