The UK doesn't really need "secret launch codes" from the UN. No sovereign nation currently submits itself to that kind of U.N. oversight. It's a plot device. Doctor Who is more Science Fantasy than SF (It has many of the trappings of Science, but does not attempt to maintain a hard connection to them, as SF typically does). It does not try to take itself too seriously, so, while it may reflect reality in some things, it takes liberties with both facts and science which SF would not. However, in the Doctor Who universe, the UK has held the codes for nuclear weapons since the Fourth Doctor serial Robot.