And perhaps my answer wasn't quite clear enough.
Although I've been in many radio studios, the only one where there was a continuity announcer was Bush House (BBC World Service). The continuity announcer (who was an experienced journalist) sat at a desk, working on the reports for the station's news output. At the same time he kept a careful eye on the clock and, as the end of a programme approached, donned his headphones and placed his hand on the fader in readiness for making his continuity announcement.
Apart from Radio 4, World Service and possibly some BBC TV output, 'live' continuity announcers no longer exist anyway. Many continuity announcements (e.g. those on BBC Radio 7) are pre-recorded, so that the programme links which will go out over several hours are actually put together in space of a few minutes.
Chris