When you say you are a mechanical engineer, do you mean an aircraft engineer ? If no, then what type of engineering are you experienced in ?
I began my career at Monarch Airlines as an apprentice back in the late 80's. I spent 18 years between Monarch and American at LHR as a technician, but always wanted to try flying....
So, I took 6 months unpaid from AA and got all of my FAA ratings up to CPL/ME/IR. I then flew for American Eagle for a year, which was a pretty miserable experience ! I left AE and went back to a maintenance position for 3 years working on corporate aircraft, this is where I finally got a lead to my current employer. I fly as a co-pilot / technician for a corporate operator in a Falcon. It is by far the best job that I have ever had, I get the best of both worlds in flying and maintaining aircraft and really enjoy it. This all came about by putting myself out there and being patient. It took some time to end up where I want to be, but I did a fair bit of networking and it paid off. My job was never advertised, I just met the right person at the right time. My maintenance background really sealed the deal for me. It is a very valuable qualification to have working for a small operator and I am very much appreciated by the people I work for, which is something that is hard to find these days.....
I don't know if you will find an "apprenticeship" that combines both flying and engineering, but as you have just read, you do have other options. Good luck in your hunt for a position. just remember this one word, NETWORK !