If multiple paladins (say an Oath of the Ancients paladin and an Oath of Devotion paladin) are in close proximity, do the effects of Aura of Devotion and Aura of Warding both apply?

Or does their overlapping mean you have to choose which is in effect?

It's been clarified that the effects of 2 Auras of Protection don't stack, but I can't find anything about the answer to whether the effects of 2 different auras stack.

Do different auras' effects stack?

Yes, the two different Paladin auras stack

do the effects of Aura of Devotion and Aura of Warding both apply

They don't have the same name, so they both apply to whomever is within the aura(s) area of effect. The Paladin's aura is a class feature (see below)

There was an errata to the DMG that says this:

Chapter 8

Combining Game Effects (p. 252).

This is a new subsection at
the end of the “Combat” section: Different game features can affect
a target at the same time. But when two or more game features have
the same name, only the effects of one of them—the most potent
one—apply while the durations of the effects overlap. For example, if
a target is ignited by a fire elemental’s Fire Form trait, the ongoing
fire damage doesn’t increase if the burning target is subjected to
that trait again. Game features include spells, class features, feats,
racial traits, monster abilities, and magic items.

The bold and italics I added for emphasis.

Yes, the benefits of different features can stack on the same target.

The relevant rule is "Combining Game Effects" from DMG p. 252 (originally added in the DMG errata)

Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items. See the related rule in the “Combining Magical Effects” section of chapter 10 in the Player’s Handbook.

The related rule from the PHB/basic rules, "Combining Magical Effects", only mentions spells:

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect--such as the highest bonus--from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.

For example, if two clerics cast bless on the same target, that character gains the spell's benefit only once; he or she doesn't get to roll two bonus dice.

Since the two auras are different features with different names, the benefits of both auras apply to anyone within the area of both.