Films and TV shows featuring United Nations agencies were quite common in the mid-Sixties/early Seventies, particularly when there was an element of SF/Fantasy in the premise. It allowed the characters to travel the world and face larger-than-life international evil criminal geniuses/conspiracies/crime organisations, without getting too tied up with real-world politics.
The Doctor Who episode you mention is a case of this: UNIT (United Nations Intelligence Taskforce) made its first appearence in the series in the 60s, and was a fixture of stories right through the early 70s. It made some sense at the time – if the world really was under constant attack by extraterrestials, then you might expect some sort of semi-covert international military force to be set up to combat it. What complicates things somewhat is that those stories were set “sometime in the near future” – when exactly was a little vague and inconsistant, but it meant that they could include details that might (in a light SF setting) plausibly happen in the near future – video phones, super computers, a greater degree of international cooperation on defence matters, etc. And, in one story, the information that the nuclear weapons based in the UK were under the control of the UN.
Now, in 2005, we’re some years beyond the “near future” that story was set in, and in the real world, UK nuclear weapons are not, and never have been (as far as we know) under the control of the UN. But, for whatever reason, Russell T. Davies chose to maintain continuity on that point with the old series.