Integrated is probably the quicker way.
But to get a job at say Easyjet though CTC, you don't have to be one of their start-to-finish cadets.
You can do modular with CTC, so you go to them once you have a PPL and enough hours to do the CPL, and the ATPL theory, and then you do the CPL/IR/MCC with them.
Or you can apply to CTC to do the MCC once you already have your licence, and can get into their infamous hold pool that way.
I've heard from those in the CTC bubble that the current "waiting list" for CTC and "non-CTC" in their hold pool is circa 4-6 months from entering the pool (i.e. finishing MCC) to type rating start date... which includes waiting for airline selection, actually passing selection, then waiting for type rating to start.
Seems like they can't get enough through at the moment