Given this throw-away line in the Serenity Pilot, it would appear the Londinium is a (possibly backwards) planet ruled by a monarchy.
Or it's a fairy-tale reference. Either way.
Fox had a Firefly website (long gone) with clips, interviews etc about Firefly.
In one of these clips Joss Whedon talks briefly about the core, he says: "There is two major planets in the core.. erm, the central planets are Sihnon which is basically China and Londinium which is basically America."
With this information, it seems most probable that Mal's remark about wanting to be Londinium's king was him merely being flippant, and that it is Alliance ruled, not monarchic.
FireflyFans.net thankfully mirrored the Fox site for posterity. The interview is here.
In River's classroom scene of the movie "Serenity", Londinium is the second planet listed on the screen to the left of the teacher.