There are three schemes which are offered under the 'Cadetship' title and I believe all fall under what was once a single company/group of companies.
Atlantic
Highland
RVL
All advertise the schemes on their websites but finding out about when they are recruiting seems to be about luck and magic as they are not well advertised.
I have contacted all three on various occasions over the last few years and was even interviewed at Atlantic for an Ops job. But I have yet to be in the right place at the right time to get a Cadetship application in!
As for being overqualified and aiming higher. Sponsorship-wise unless you're a national of a country which sponsors for its flag carrier (or lucky enough to be the best on the planet for Etihad!) then there is no other genuine schemes to aim for, higher or otherwise.
I am a masters of Physics graduate with 2 years professional experience in another industry. I would be unlikely to seriously struggle with the academic side, and would still apply to this scheme if I could. So that's the kind of competition you'll be up against from a qualifications point of view. But it will ultimately be about much more than that separating the selected from the rejected.
Good luck all.