Going by how alignments are defined in my campaign setting/world...and they are defined and used and an important part of your character. Alignment -and the forces of Good, Evil, Order and Chaos- does [conceivably] have consequences in your character's life...your paladinhood is defined by your oath. You have to take that oath seriously. Take it/make it as part of yourself. Your word, your promise, your OATH is to this cause. If you are not capable of the discipline needed to maintain and enforce your word/promise/oath to your cause of choice...then how are you expecting to be -or continue to be- a paladin?
Paladinhood, at its core, demands Lawfulness.
A paladin, not just needs but, holds at the core of his/her being a conviction toward Order. There must be the discipline to maintain your convictions and follow your chosen path [when it will, presumably, not be so easy a lot of the time]. There is an eye toward civilization and, well, law, and the furthering/maintaining of it (the devotion to that Oath you hold so highly hold: be that through individual power, the forces of "Good", "the light", the king, the nation, the order, Truth, justice, etc...)...regardless of whatever other concerns create their particular cause/order/deity/other focus of their abilities.
That in mind. I can see, in 5e, a breakdown like this:
Devotion: obviously intended as the Lawful Good option. Lawful Neutral and Lawful Evil are easily doable with minor refluffing.
Ancients: obviously intended to be the "nature=neutral" option. I would submit that this is, primarily, the "Good" Oath. You are not so concerned about the "Law" as the "Light". Put another way, the Spirit of "Law"[of the multiverse] vs. the Letter of a Devotion Oath. You are still doing good, making things better, making things enjoyable. Yes. Protecting the good/innocent. Yes. Defending and building up the weak. Helping others and, thusly, maintaining/creating the beauty of life through that assistance.
In a 5e game I would run...I would insist Oath of Ancients Paladins be "Good". I would probably prefer, most commonly, Lawful Good and would certainly allow/see Neutral Good...and, in rare cases, a Chaotic Good (for the player that insists) who is kind of clinging to their paladinhood by a thread...and the player would be aware of it. [I would further suppose that a "fallen" Chaotic Good Ancients Paladin of any significant level would almost CERTAINLY be noticed/picked up by some Archfey and transition from the loss of their paladin abilities into a Fey Blade-pacted Warlock.]
True Neutral doesn't really work here, to my mind/game/world anyway. The True Neutral is no more concerned by the Light as they are by the Shadow. The BALANCE of the two is what's important. That is not to say a True Neutral character can not be as convicted/devout as a Lawful character (see below), far from it. I still insist, in my games/setting, that Druids are [and can only be] True Neutral. But that does not work for a class whose Oath is all about bettering, protecting, fighting for life and beauty.
Vengeance: This is obviously meant to be the Batman figure. The guys who create and plays by their own rules...but they are still rules! And those rules are very important. There is still that word/promise/oath and so a closeness/appreciation for the law/order/discipline that maintains such ideals. There is still the conviction to the cause [of vengeance]. The individual [selfish?] nature of such a cause, I would submit this is intended to be the subclass for the Lawful Evil character. I can also see Lawful Neutral, doling/seeking out vengeance for others, as well as themselves...like Nemesis of Greek mythology or other "Furies" type characters who roam around looking to carry out "justice", as they see it. Lawful Good is a tough sell here. I might be inclined to go as far as True Neutral or Chaotic Good...again, for a PC/player who asked/insisted. I could see it.
The leaves the Oathbreaker [which I would never permit in a game, actually] for what they are...Oath breakers. Their "Lawfulness" is shot to hell. They are the "Anti-Paladin" who, unlike the original class of this name, are the opponents of LAW...not only [or necessarily] good. They are champions of Chaos. They did not have "what it takes", by purposeful decision or personal failing, to maintain their word/promise. They did not value Order and Law enough (if not actively oppose it) to keep their original status. So, it is obvious to me, this is your option for Chaotic characters...Evil probable/most common...but I could conceivably see a Chaotic Good Oathbreaker. Perhaps even a remorseful Neutral, who lost their faith/way on the path of Law/Order from some dire/no-win situation, but still does not wish/follow/promote evil. Both CG or N would take some refluffing of Oathbreaker features, be that as it may... Chaotic Neutral is just...a step too far. Not a paladin. No matter how you slice it [in my definitions of alignment], there is no good way to justify a Chaotic Neutral Paladin.
SO, in summary, ordered for commonality/intentions [for me game], it goes like this:
Devotions: Lawful Good. Lawful Neutral. Lawful Evil.
Ancients: Lawful Good. Neutral Good. Chaotic Good. [maybe Lawful Neutral].
Vengence: Lawful Evil. Lawful Neutral. Chaotic Good. [maybe True Neutral].
Oathbreaker: Chaotic Evil. [maybe Chaotic Good or True Neutral.]
So...you can't be ANY alignment and be a paladin in my games. But most of them are an option. I prefer/try to insist they consider being some flavor of Lawful first...but if a character concept ["I wanna be Batman!"] doesn't allow for that...that's up to the player.