You can start IR training before the ATPLs are complete and prior to the CPL so long as you have a multi and night rating. You just won't be able to do the IR test until the ATPLs are complete. As mentioned above, you must have the hours and ATPLs before starting CPL training.
If you do the CPL first you get 5 hours off of the IR. If you do the IR first you get 10 hours off of the CPL.
See LASORS for details.