I've completed the Imperium campaign on Hard difficulty. I even have the achievements for it. Yet, the game refuses to let me start a Necron or Tyranid campaign because it thinks I haven't completed the prologue and imperium tutorials.

Hey @badhras
You can change this in the "game.ini" file.
If you open that file and find the following:

bHasPlayedPrologue=True
bHasPlayedImpTuto=False

Make sure both lines are set to "True". Hope this helps, please let us know if it works, thanks!