Is it only Apple TV (and not on iPhone for example)? If it's Apple-wide I can imagine it but I'm curious what is the possible reason that BBC can't support having a separate subtitle renderer?
CC is a bit of an Americanism, perhaps it is running more directly though some Apple playback code that is more particular about subtitle formats?
I suspect the ultimate case is probably low usage statistics not making it a priority. The icon / BBC logo didn't update at the same time as the iOS version, so I suspect its a separate codebase for some reason?
I’m guessing there are some complications hooking Apple TV’s native subtitles APIs to relevant web APIs, and low usage statistics doesn’t help prioritising fixing the issue. Although the rumour is that they are working on a completely new Apple TV app.