On radio, it seems the first swear word (that we'd still consider to be a swear word) was 'arse', said by a muisic hall artiste called Hector Thaxter. Unfortunately I can't find any information about the context.

The same site I found that snippet (http://www.laughingpoliceman.com/swear.htm) also reports that in 1957 a teddy boy interviewed live on BBC news said 'piss'. This might be the first TV sweearing, although annoyingly the site doesn't mention whether it was TV or radio news.

The really famous swear on TV was in 1965, when theatre critic Kenneth Tynan said the f-word on a satirical show called BBC3. I think he said something along the lines of "No one would be offended by the word 'f***' nowadays," a point amply proved by the huge furore that sprang up in the wake of his comment, which included calls from one MP that he be hanged. Mrs Mary Whitehouse even wrote to the Queen suggesting that Tynan needed to have his bottom smacked, although why she thought the Queen would be interested remains mysterious, and Tynan would have rather enjoyed having his bottom smacked, especially by the Queen. Who wouldn't?