My question is: Do people who train part time, doing each licence as they go along, have just as much chance of becoming an airline pilot as those that are able to do a full-time, integrated programme?
The simple answer is that very, very, few airlines are interested in people with fresh licences and two hundred odd hours. They aren't now and they never have been. The exception to this is those airlines that have cadet programmes as part of their recruitment intake. Most of those airlines contract with one or more training schools to provide the basic and intermediate portions of that
ab-initio training. The successful graduates then complete their advanced training with the partner airline. Of course the basic and intermediate training is full time, integrated and specific to a small handful of training schools.
Beyond these cadet programmes, you can work your way up through career advancement. Many people over the years have worked as flying instructors and in general aviation, and then made their way up through the "stepping stone" jobs and second and third tier commercial operators until they have the broad brush experience that the first tier airlines generally stipulate
as a minimum for direct entry recruitment. Those figures vary from company to company but again, broadly you are looking at around 2000-3000 hours with 500+ hours of turbine time (turboprop or jet.)
The reality that has evolved over the last 15 years and continues to increase in significance, is that there are very large numbers of pilots with CPL's (or fATPL's) these days. In the last decade the basic requirement for CPL issue in the UK (in most cases) dropped from 700 hours to around 250 hours. The reasons for this I have explained many times and a quick history search should save me having to repeat them again. However, the reality is that the floodgates were opened to thousands and thousands of prospective CPL aspirants. Coupled with a contraction in the availability of "stepping stone" jobs, this left whole swathes of fresh CPL/IR holders in a vary crowded arena holding very perishable licences. The results of that you can read in the many thousands of threads on the subject.
At around the same time that all these licence regulatory changes were happening, two other notable things happened. The first was the introduction of age legislation that quickly served to removed the early retirement profile for many established airline pilots. This gave many pilots (and their employers
the airlines,) another 5-10 years of breathing space. The second event was the expansion of the airline cadet programmes. The result was, (coupled with a long global economic downturn,) that a huge swathe of airline entry level potential, that previously would have been satisfied by the 2000-3000 hour career advancers, was now being satisfied by the maturing cadet programmes. A cadet will only take 36 months to be at the 2000 hour+ level and of course almost all of that time is turbine time. Not only turbine time, but type experience and airline experience in exactly the measure the airline wants.
This has resulted in significantly reduced entry opportunities for career advancers and career changers (ex-military pilots moving to a civil environment, and experienced pilots abroad, looking to return to the UK, by way of just two examples.) For the fresh licence holder outside of these programmes that has resulted in pressure from below, and pressure from above, culminating in a great many people being squeezed out of the marketplace.
So you can see why a simple answer to your question is not as straightforward as you might hope. With a fresh CPL/IR most airlines with cadet programmes are simply not going to entertain you unless it is through their cadet route. Many others will simply say come back when you have enough of the right experience, and a very few may let you compete for their entry level recruitment in what is an extremely crowded and corpse laden field. The "stepping stones" are still out there, but the incumbents (they will be along shortly!) are hanging on to them for grim death, and almost everybody in that crowded field is looking for a way out.