You should get in touch with my old mob at BAE Systems in Warton, Lancashire. Since they moved the Harrier up from Dunsfold it is the only military flight test function outside of the services. In the past we have had work experience types - mostly from the local schools. They generally get some exposure to a trial or two that will be underway at the time (sit in on briefings, get involved in aircraft preparations etc) one or two have even had day trips on the company communications aircraft down to Farnborough & back.
Give the swicthboard a call on 01772 633333 and ask for Human Resources. Then ask for some one that deals with work experience placements in the Technical division. Flight Test is part of that.
If that doesnt work, try AVRO over at Woodford - sorry no phone number at hand!
Other than those two, the only way you would get into flight testing would be through the ATC and asking for a stint at DERA or New DERA or whatever they are this week. Not sure on their policy. Best thing would be through the local ATC.
Good luck!