A few years ago I wrote to EVERY airlines address i could find operating out of heathrow and gatwick and got a abrupt NO or no response at all from all of them I'm afraid. Same answer from the ground handling companies too. Security is just way too tight at the bigger airports now.
However I did have a very enjoyable week with Flybe. in exeter in 2003. It seems to really depend on what staff are around but they'll have you spending time all over their base from operations to the matainence hangars. If you're real lucky, and a certian flybe guy pulls the right strings the airport operators may take you round their bits of the place for a day too (the tower etc..)
They take on one person a month but you'll have to write to them at least 6months ahead to get booked in, and hopefully have a old relative nearby to stay with...