Dom, BA have component overhaul workshops in Feltham, some distance from the airport. The guys who work there do not as a rule work on aircraft and are not licensed engineers under Part 66. BA recently took some functions back in house having formerly farmed them out, so it's unlikely to be outsourced, but it's not a glamorous job or (I believe) particularly well paid, based on pressure to get components turned around and out the door as quickly as possible. To this extent they are quite unusual as most airlines these days outsource their components to specialist companies.
A career in component repair is fine if that's what you're looking for but it won't lead you into aircraft engineering and the CAA do not count time working on components to be experience working on aircraft for the purposes of the licence. You may get a foot in the door with respect to internal vacancies but that's about the only 'plus'.
There have been many threads on here about how to start a career as a licensed engineer and all seem to come up against the same catch-22. You can't get a job without experience, and you can't get experience without a job. However time is on your side and you might like to consider asking people for Saturday jobs (if necessary unpaid) at flying clubs so you can gain experience and demonstrate commitment for the next stage, which is to apply for apprenticeships. Do some research on which companies may be prepared to offer these. Start with Monarch, Inflite, ATC Lasham, Thompson or even BA. This list is not comprehensive, do some homework! Expect a lot of competition as there are few places and much demand. If you do not get an apprenticeship don't despair, you may be able to continue to learn via the flying club route which will give you enough hands-on experience to apply to agencies as a contractor. Once they have you on their books you can start working on aircraft and logging hours towards your engineer's experience requirements. You would have to find the costs of your exams/tuition yourself this way but TBH the first rung on the ladder is hardest. If you can get working on aircraft the rest is just money.
There is always the option to join the RAF of course, plenty of experience and tuition if you don't mind getting shouted/shot at
Have a look at some of these links which may be useful background reading for you, in particular the ELGD.
Regulations structure
Download and read 2042/2003 which are the basic regulations covering Part 66 (section A for companies and individuals, section B for the regulators), and the relevant AMC (acceptable means of compliance). It's good bedtime reading if you can't sleep. These regulations do get superseded and added to periodically but it's good to know the basics.
Documents Categories | Personnel Licensing | Safety Regulation
All you need to know about engineer licensing under the CAA.
Note there is a very strong possibility you may only find a placement some distance from home so you may find yourself having to travel or live away from the olds whether you like it or not. Get through your A levels while working Saturdays, get a vehicle and you will have many more options.
Good luck!