Lump Jockey
You are right about the construction of the frozen ATPL (PPL, more flying, groundschool, CPL and IR), but that is the modular course. The integrated course has all the training completed in a single, long block, mixing flying training and groundschool and with no intermediate qualifications, just stages of training with proficiency checks. This gives you only about 170 hours flight training compared with a modular course at around 230 hours for most people, but due to the extra resources the school has to show to gain approvals it costs at least £10,000 more to go through an integrated course.
The integrated is also less flexible, as they usually require payment up front and the rules make it hard to change course if you find that you don't like the training provider. You also need to spend the whole year and do the flying and groundschool in the same place or perhaps part of it abroad in a specific location, when you might prefer to be close to home for one section but feel that another section is better taught elsewhere. Each module of a modular course can be taken wherever you prefer.
BA do say they won't take anyone straight out of a modular course, and nor will Emirates but as neither takes many low-hour, unsponsored pilots I am not sure how much benefit the integrated student gets. At your age especially that should not be a consideration!
If you have any specific questions that you feel that I (as an employee of a school that runs all the modular courses but no integrated training) can answer without bias then feel free to send me a private message or ask directly here. I have completed the whole modular training and I teach PPL flying and ATPL groundschool, so know the system fairly well.
If you want to know more about integrated training I am sure that the instructor who posts as Oxford Blue would answer any questions - he works for Oxford.