There are two version of the BBC iPlayer app for iOS, there is the UK version which has content from the last week of BBC TV and Radio and the global version has a selection of BBC content including Doctor Who and Top Gear.
The Global version has been updated to include Airplay streaming so you can watch iPlayer content on your Apple TV. Strangely the UK version of the app doesn’t do video streaming from Airplay and unlike the global version doesn’t support downloading for offline playback. Lets hope the UK version catches up soon.
What would be nice is if the BBC added iPlayer directly in to the Apple TV