What does Detect [Alignment] reveal about a creature that is possessed? Does it reveal the presence of the base creature's alignment, the possessing creature's alignment, or both?
I think the answer would depend on the alignment of the creature being possessed and the alignment of the possessing creature. Looking at the spell Detect Evil from the PHB:
So if an evil creature is being possessed by another evil creature, the caster would be able to detect that there are 2 evil aura's on the creature. Note that the caster doesn't know the source of the aura; hence they couldn't tell if it is possession, an evil magical item, or an evil spell on the creature.
If a good creature is being possessed by an evil creature, the caster would be able to detect that there is 1 evil aura on the creature. Again, since the source of the evil is not known there is no way to tell if it's a possession, an evil magical item, an evil spell, or if the creature actually is evil... though most players jump to the conclusion that it's an evil creature, which can make a great adventure hook.
Possession is usually described in fiction as someone's mind (or a piece of it) entering one's body and taking control over it from the rightful owner; contrast with domination and other forms of mind control, where the controller's mind stays where it is but sends instructions.
Detect Evil in the PHB says that you can "sense the presence of evil", and the other parts of the description all refer to evil auras (basically to detect strength and number of evil presences). So I'd think that if one creature possessed another, then both of them would show up for Detect [Alignment] (subject to their actual alignments) since both of them are present in some form.