Tough one this...
The crux seems to be that airlines are now run by managers whose only focus is cost, and that means no-one does apprenticeships anymore because they cost money.
This has led to the ludicrous situation so aptly described by the above posts - to become an aircraft engineer you have to go to uni, but following that, the airlines don't want you because you have no hands on experience or history. Then the industry bleats on about a lack of qualified, experienced signatories..
As stated above, the most effective way in is to start at the bottom and work your way up, but that is of little comfort to someone who has just invested the last three years in a mountain of student debt and a shiny new degree certificate. It may simply be the case that you will have to start as a basic mechanic, greasing undercarriages and cleaning windows, and work through technician up to license holder.
I was lucky enough to be awarded a place on one of the last 'proper' apprenticeships run by BA in the early 90's, where one of the real advantages was that by the time I was allowed to actually do anything on an aircraft, I already had a three year long proven track record - my employer already knew me and my strengths and weaknesses. It is a bit of a risk for someone to take on an engineer with no proven experience, but that would appear to be a catch 22 situation entirely of the industries own making, and unfortunately, as a by product, you (and other new engineers) are suffering as a result.
With respect to my esteemed colleagues, I'm not sure why having a degree should preclude you from line work - it at least demonstrates that you have a brain and are willing to work towards a goal, but I will also back them wholeheartedly in the suggestion that that is only half the battle... an academic accreditation is not the same as doing real work, on a live aircraft at 3am, in the snow, without the proper equipment, with fingers so numb you can't feel what you are doing and an airline breathing down your back insisting that they want to fly the plane in 2 hours time.
However, that still doesn't mean that deep down you aren't capable of such self discipline, but in this business there is no substitute for experience and you DO have to prove yourself I'm afraid.
I think, as with most of the important jobs in aviation, (Pilot, Engineer, ATC etc...) the only route to success is sheer hard work and determination - after all, you don't get to Captain a Jumbo without slogging about in a Cessna first, degree or not.
I wish you the best of luck, I hope my post isn't too negative, and remember that if you really want to do it, with a bit of effort you probably can, assuming that on a basic level you actually have the ability...
