LGW,
I managed to visit Gatwick Tower a couple of years ago with my son (then aged 13). It was as a guest of the Airfield Management though so that may have made a difference. However, it could also have been that since he was 'accompanied' by a guardian it was possible to issue the pass.
Also, as the Rev indicated, it may be an insurance issue. I'm not sure but I have a feeling that one of the pieces of paper I signed at Security may have contained a clause to the effect that I, as legal guardian, took responsibility for my son while airside.
As an aside there are similar regulations regarding access to the pitlane at race tracks - no under-16's allowed - probably for similar insurance reasons.
Hope you get it sorted out though.