Not an awful lot in Berkshire, but I think that there's some kind of maintenance setup at Brimpton (a little E of Newbury), try phoning Sylmar Aviation on 01635-866066.
You could try going over to White Waltham (between Reading and Maidenhead) and asking around - it's a big place with not only the main airfield and flying club, but quite a lot of small aviation businesses there.
Ditto Thruxton, which is a little way West of Andover.
There's a microlight school at Chiltern Park, which is about 4m NW of Reading. I've no idea if they are big enough to offer any summer work, but you can't do yourself any harm by trying. Same applies to the microlight school at Clench Common, just South of Clench common. The problem with these for you as a 16YO, is they'll probably both be almost impossible to get to without a car or aeroplane.
But basically, go around the houses, talk to them in person, be polite and presentable, and honest about what you're after - the worst that can happen is that they say no.
And best of luck,
G