You might not like this reply, but I've got to ask. If you wanted to work "hands-on" why did you go down the University route to start with?
The problem you've got [as I see it, from the point of view of an Engineer who has been in Aircraft Maintenance for 30+ years] is that many potential employers will probably see you as over-qualified for a Mechanic job. They will probably think you will either get bored or will not want to stay on the "shop-floor" for very long with your qualifications, so the money they spend on you in terms of training/courses etc will not give them a good return on their investment in you.
Also be aware that many employers are trying to "de-skill" the job as much as possible, with many major airlines putting pressure on EASA to do away with the "expensive" [as they see it] licensing system and go to a much cheaper [for them] company approval type system, so long term I do not think things are going to get better on the shop floor end.
As you state, most graduate schemes are aimed at recruiting staff that will go into Technical support roles rather than hands on type work.
If you really do want to go down the Mechanic route you could always try going in as an Apprentice with an Employer. It might sound like a backward step but I know we have had Apprentices start in the past where I work [a certain
Big
Airline
] that have had degrees.
So you might want to try BA for either Apprentice or Mechanic jobs [There were due to be more intakes in the coming financial year, which in BA now runs Jan-Dec as the holding company IAG is registered in Spain] but there is another financial review about Engineering costs going on at present, so I think it's on hold. Remember BA employs Engineers not only at Heathrow but also at the Major overhaul bases in Glasgow and Cardiff.
Good luck with your job search and I hope I didn't sound too "downbeat". I wouldn't want to discourage anyone who genuinely wants to work "on the tools" nowadays as many don't want to get their hands dirty any more!