As far as I know you can start the CPL without having passed all the exams; however, you must have passed them before you take the skill test. You must though have the correct number of hours before you start the CPL training.
However, a CPL course is usually quite intensive and most people do it pretty much full-time. There's no point putting extra pressure on yourself to have to study and take exams as well plus, if you do fail any and your skill test is consequently delayed, you'll get rusty and then you'll need refresher training which would nd up more costly.
So, there is very little point in starting the flying; just concentrate on one thing at a time.
Cheers
Whirls