The wedding in the first part is shown to drag on forever, starting hours later than planned and lasting well into the next morning, and is a miserable experience for all involved. When the bride, exhausted and depressed, is finally able to get some sleep, she dreams the second half of the movie.
While Lars Von Trier freely states they made no attempt to get the astrophysics correct, they did a decent enough job to make Melancholia believable within "dream logic" (i.e. You accept it during the dream; only after waking up are you able to rationalize that the content of the dream was impossible.)
The entirety of the film takes place at a remote, wealthy estate. We never hear or see any news reports about what is almost certainly the single biggest event in recorded human history. Given what that event is, the rest of the world must be in utter chaos.
Justine's wedding shows a more prosaic sense of how depression can wrack an entire family, but the second half of the film is more an abstract reflection on the aftermath of how depression ravages a family. When Claire does a search for "melancholia," most of the hits are simply for depression. In a way, all of the characters are simply caricatures of different styles of melancholy: Justine is depressed, inward-turned, and lethargic, while her sister Claire is more anxious and frenetic when under crisis.
This reading of Claire also explains her husband John's behavior: he tends to be dismissive of Justine's depression, of Claire's fears. He insists that Claire not surf the internet — much like a beleaguered spouse might insist that their hypochondriac partner not surf the web — in a manner both nurturing and condescending. However, when he realizes that the crisis is inevitable, his rational framework falls apart, and he commits suicide.
As the Star-Child, Bowman returns to Earth to discover that the interference of the Monoliths has utterly ruptured his world's timeline so that by 2001 mankind has not reached the technological brilliance he was accustomed to; in fact, by 2012 they're still as superstitious and unenlightened as ever. Angry that he is the only enlightened human left, he assumes the form of an immense planet and rams the Earth, destroying what he sees as a doomed species.
But this doesnt happen, as it is clearly stated in-universe the defense forces of the Imperium have been severely depleted over the millennia (just as its said in Dredd(2015) the judges are so far and few between they can only protect as small as 10 to 15 percent of Mega City One) so either the Imperium had received the message and only didn't have the resources or troops to aid them in their plight, or just as likely the hive world probably wasn't vital to overall needs so they decided to sacrifice the planet to stretch imperial resources.
So instead of interstellar saviours such as squads of Space Marines or Exterminatus wielding Imperial Knights they get Nothing. So all that occurs is probably some of the population evacuated into underground bunkers in the false hope of safety from the initial blast and thousands (if not millions) of colonists desperately sending countless prayers to the Emperor and the Omnisiah for survival. With the remaining populace rioting in the streets.
Nothing of the outside world is really ever mentioned throughout the film and most of the story intentionally invokes Grimdark till its coming out of the characters ears. In fact this kind of stuff happens a lot so this could even be seen as a simple slice of life story within the 40k universe.