I'm from Newcastle and have an ambition to become an airline pilot in the future. I wrote to Newcastle Airport around this time in 2006 and received a standard 'no, due to security measures'.
However, I wasn't satisfied with the no, especially as it was just from their customer service/marketing department.
I therefore wrote to the Air Traffic Control director at the airport and managed to gain a 2 afternoons work experience.
However, as previously stated this was in 2006. The industry was expanding and security wasn't as tight then. With the recent increase in terrorism threat and the government wanting to step up on security at airports, I'd imagine that it will be very difficult to get work experience at the airport! It's also worth mentioning that in 2006 the old control tower was used closer to the terminal building. Today, the new tower is located at the other end of the airfield so it may be harder to get there because of security.
Good luck!
EDIT: Also, I wrote to just about every airline and they were a no too due to security. Flybe said they do offer work experiance but only at the hub in Exeter, which is often booked months in advance.