This is a "Wild Mass Guess" entry, where we pull out all the sanity stops on theorizing. The regular entry on this topic is elsewhere. Please see this programme note.
The Department is actually Department C19
Department C19 is mentioned on-screen in the Doctor Who serial "Time Flight", where it just seemed to be the UK government's UNIT liason. The Doctor Who Expanded Universe makes it a deeply sinister group involved in things UNIT couldn't imagine and particularly interested in controlling alien tech. They were supposedly shut down in the nineties, but apparently weren't, since they appear in works set after that. Clearly, C19 survives until the time of K9 and, having taken over every other arm of the government, becomes known simply as the Department.