I qualified at a 'local' flying school, at the peak of the job market, and got an airline job within three months. My airline now has sponsored and mentored cadet pilots, who trained at 'highly recognised schools', working as cabin crew because they qualified at the wrong time and there are no pilot jobs for them.
When you qualify is much, much more important than where you qualify. Don't be taken in by the schools' marketing spin - this is a terrible time to train. You're 21 so not exactly old - review the situation in 2 or 3 years time, but whatever you do, don't believe any school that tells you this is great time to train.