No, they just executed their announcer for treason just for what he said on radio and accepting German citizenship (and it was not even clear he was legally a British subject to begin with...). Also jamming it back then wouldn't have been that easy and worthwhile also the Nazis occasionally allowed POWs to send messages to their families back home.

Btw the scale of censorship and other measures restricting the freedom of speech in Allied states during WW2 (or WW1 for that matter) were was extreme by modern standards.