IIRC, all should overlap.
You can get a CPL without an instrument rating, so the minimum instrument requirements for CPL are geared towards a non-IR pilot who needs some instrument training just in case he inadvertently goes VFR-into-IMC.
That's for part 61, unless otherwise specified, for the most part any training done at a given certificate level counts towards aeronautical experience requirements for the next certificate. As a private pilot, you meet the "prerequisite" to perform CPL training.
That's for training, ie specified activities with or without a CFI. For flight time, that can be done at any time. Ie a PPL with 1000 hours will not need to complete an additional 1500 hours to get an ATP after he gets his CPL. He can get the 1500 at any time student, PPL or CPL (although it's not really the intent that you solo for 1500 hours as a student pilot, and the FSDO would have words with the CFI in that case).
If you do CPL under part 41, you'll probably have to the syllabus in order although you might be able to get partial credit for some previous flight time.