Three points.. The obvious way to contact ATC by telephone is to ring the airport and ask the PBX operator for ATC or the SATCO.. In the latter case you'll probably get his secretary unless it's a tin-pot place where the SATCO has to wear a hat. You could use this method to find the name of the SATCO, or Officer-in-Charge, or whatever he/she is called. For large airports the SATCO probably has little to do with direct recruting and you'd need to contact the Personel Dept (or by whatever name it's known).
Two.. I would suggest writing but be warned that when I tried (a hundred years ago) the reply rate was about one in ten million.
Three.. Why have you removed NATS from the equation? Despite the opinions you might read on here, NATS are good employers and staff enjoy very good conditions of service compared to elsewhere.. I know . .... some may not agree so you have to reach your own conclusions.
Good luck...