I thought I would post an update of the replies I got so that others can see what the situation is like with work experience placements these days:
NATS – Not in the towers (not practical), Swanwick ok – placement subject to interview, and all were allocated.
For 2007, they require a CV, passport sized photograph and a letter of recommendation from your school/college by 30th November 2006. (For contact details feel free to PM me).
GB Airways – Supports a number of local schools, but all were allocated
Excel Airways – First letter said they were currently in the process of constructing a work experience program and that they will get back to me, and the second letter said they were unable to assist in my request at this time
MyTravel – Manchester only
Flybe – Exeter only
Thomas Cook– No
Flyjet – No
British Airways – No (says on its website)
Astraeus – No reply
BAA – No reply
I found that it should be possible as long as you apply as early as possible, and try as many companies as possible (I didn't think about trying the handling agents and missed out some airlines) and are flexible as to where to apply.