The Hellknights are technically independent of any government oversight; they're somewhere in between vigilantes and independent military contractors. That being said, the Hellknight orders are, above all else, extremely lawful in outlook. I'm sure that the ones that go around capturing criminals and the like observe some kind of due process, but I suspect that they examine the evidence *before* they go after somebody, so once you're caught I'd guess that you'd be "processed" fairly quickly.
As for Cheliax, they absolutely have regular law enforcement; as I recall, it's mostly down to various city guard organizations to carry out arrests, though army units probably spend a lot of time rooting out bandits and other criminals outside the cities. One notable feature of the system is that bribery has basically been institutionalized; for most crimes it's acceptable for the arresting officer to accept an appropriate 'fine' in lieu of other punishment, presumably assuming that their superiors get the appropriate kickbacks.
As far as I can tell, the courts in most parts of the country are more or less run by the church of Asmodeus, so trials likely swift and more than a little cruel. Executions are a popular form of public entertainment.