Hi. It's almost impossible to get work experience because ATC staff are highly trained. Controllers hold licences like pilots which involve several years of extensive training. Also, only employees are covered by company insurance. Swanwick would almost certainly be out of the question but I can't say for small aerodromes where you might get work as some sort of ATC clerk, but very few places would take you on for a short period.
If you want to get into ATC why not apply to NATS for a cadetship?