[whswhs]

I hadn't realized that this was a Doctor Who question, as I've never seen an episode of Doctor Who. But looking at the question generally, I'm going to say it's not something you would do with any of the methods in GURPS Social Engineering. They simply don't provide a means to such extreme results.

I would tend to look at that sort of thing primarily as a political superpower, though I'm not sure if GURPS provides the requisite advantages to define such a superpower straightforwardly.

Alternatively, it's sort of like the application of one of the esoteric skills that cinematic martial artists use as a substitute for superpowers. There really aren't suitable skills for that, either. One of my Pyramid articles provided the concept of "social arts" that could include cinematic techniques, and it might be possible to define a cinematic technique based off of a realistic skill—possibly a political analog of Breaking Blow. Alternatively, it could find a home in a weird science discipline such as I discussed in GURPS Powers: The Weird: I suggested Memetics as a possible such discipline, and predictive psychohistory in the style of the Foundation novels, Flynn's In the Country of the Blind, or Kingsbury's Psychohistorical Crisis could be another.

If we're going to go that way, I can see three possible base skills: Politics (for conventional political campaigns), Psychology (for psychological warfare), or Propaganda (for rumor spreading, whispering campaigns, or the like). There's some discussion of all three in Chapter 5 of GURPS Social Engineering.

Incidentally, as was just mentioned upthread, the different meanings of "government" in British and American speech are important, making the source in Doctor Who crucial. "I could cause your control of parliament to fail and force you out of office" (which has no real American analogue; an American president is in office for the full term, barring death or impeachment or the like) is a major threat, but much less than "I could cause a revolution against your entire political system," which is what that threat means in American. (I would say "cause the fall of your state," but that's also ambiguous; "state" can mean either the political organization that dominates a society, or the whole society as a collective organization, whose "fall" would probably entail mass deaths—and that's not even taking into account that in American, "state" most often means one of the fifty regional units that constitute the American federation. Political language is full of ambiguities.)