There should be a search function for finding your favourite radio station on Apple TV. Finding BBC Radio 6 is like looking for a needle in a giant haystack.
After a quick search via Apple Support Communities, this was the best result for finding BBC Radio 1 to 6. For the people who are also searching for the same answer, the information is below copied from the forum post.
BBC Radio 1 – TOP 40/POP
BBC Radio 1 xtra – TOP 40/POP
BBC Radio 2 – ADULT CONTEMPORARY
BBC Radio 3 - CLASSICAL
BBC Radio 4 – NEWS/TALK RADIO
BBC Radio 4 LW - NEWS/TALK RADIO
BBC 5 Live – NEWS/TALK RADIO
BBC 5 Live Sports Extra – SPORTS RADIO
BBC 6 Music – ALTERNATIVE ROCK