Oaths have mechanical impact. Alignment does not.
You can be an evil paladin in 5E, but you still have to abide by your oath. It doesn't matter if "anyone's watching" or not. There's not some School Principle who's going to punish you, but only if he notices. If you aren't faithful to your oath, you lose your paladin status. Alignment doesn't enter into it, and neither does whether or not you're caught at it.
So asking the DMs out there, would you allow a lawful evil Paladin to take an Oath or the Crown, Devotion, or Ancients?
He'll behave accordingly in company at all times, but will probably "do evil things" in times he thinks he could get away with it. Obviously this character would have no qualms with lying, cheating, stealing, and killing if he thought he could get away with it.
Unless they're Vengeant of course.The devil, as they say, is in the details. I could see a LE paladin servant of an evil king, for example.
Whether anyone sees it is irrelevant. The Paladin's power comes from their own devotion. You can't hide your actions from yourself.
The paladin, ultimately, views themselves as in service to some ideal. The paladin is *not* all in it for themselves. The paladin who is primarily concerned with personal gain is always going to be in danger of breaking their oath, and having issues.
Unless they're Vengeant of course.
Just spitballing:Remember, the power now comes from a Paladin's adherence to their Oath, not from a God. How would a Paladin maintain their powers from the Oath of the Crown if they don't believe in the influence of the Crown? They literally couldn't.