As a general point, I believe that people will feel comfortable with people of a similar background. So if you are up against a panel, the majority of whom came through the sponsorship system, then they are going to be more interested in candidates from a similar system - probably integrated in this day an age. Given, that the known big names will add more familiarity; Oxford and BAe will be a good bet.
Pilots who came up the hard way (and there are more and more these days) will be most impressed by the self-improvers, the oldies, the can-do and made it happen types regardless of school or background.
Obviously this is such a gross generalisation but I cannot escape from the conclusion that if you are Oxford or BAe integrated then you have an advantage. If not regardless of integrated, modular or school then be prepared to go that extra mile.
This does not mean that Oxford/BAe students are generally better able, just generally better positioned. As a self-improver, I do not welcome this realisation.