I'm torn between playing a warlock or paladin in a homebrew campaign I'll be in. If I go Paladin, I'm not sure which of the two subclasses to choose from. Both would work with the backstory I have in mind, a human raised in the feywild. The only thing I know I'd want is the Mounted Combatant Feat because gosh darnit I want a beautiful elk to ride. And from what I heard from my DM both could be useful in this campaign. Any suggestions for your preference and a possible build with a Variant Human?

If the campaign mostly happens "in the wild", has a lot of travelling and exploring for the party, a "mounted" Oath of the Ancients might fit better. Any mounts are in no good place with lots of dungeon crawling and city life involved.
The Oath of Redemption Paladin has a nice "out of combat" feature with Emissary of Peace and I find that Oath to be a bit "more of flavour" for role play (that's why I chose that Oath for my Paladin when I could not decide between the two). Depending on how long the campaign will last, or at which level you will start, the 7th level Aura for Ancients if better (mathematically) than the Redemption Aura, while the 15th level ability is the other way round. I went with Interception fighting style and Polearm master with a halberd to stand next to my front line companions (leaving me with an attack option for my bonus attack and my reaction to either intercept damage or to attack incoming foes with reach).

Points for why both are good so you can compare and contrast. I'll only be covering up to 9th level though since that's when the majority of play will actually happen.

Redemption gets some pretty good debuff and defensive spells. Sleep is still able to be effective when you first get it. Calm Emotions can be an encounter ender. Hold Person lets people autocrit on an enemy while they also can't take actions. Hypnotic Pattern can knock out encounters if enough things fail. On the defensive side Counterspelling is a pretty important thing vs enemy spellcasters. Sanctuary can be really nice to protect squishy supportive allies or protect yourself as you run up to the enemy.
Channel Divinity options helps make you into a face (+5 is relatively huge), and you have a universally usable damage option that requires you to watch for the biggest hit from an enemy to really use it best.
Aura lets you take damage for weaker allies, which can be huge if the spellcaster was about to take so much damage their concentration was almost guaranteed to drop. Otherwise though it's sort of a trap feature, since taking damage is also bad for you.

Ancients' spell list is much more utility and control focused. Speak with Animals can get you information you might not otherwise have been able to. Ensnaring strike can be an easy way to get advantage for you and your allies for attacking an enemy. Misty Step is a bonus action teleport to help escape locations or chase down enemies. Plant Growth makes movement through a sizeable area nearly impossible.
Channel Divinity is a lot worse in this case, one option is limited to Fiends and Fey, and only one of those is considered even somewhat common for most campaigns. The second option is also pretty terrible, requiring you to be next to the enemy, costing an action, and giving them a save every turn.
The Aura is where it's at though. Automatic resistance against damage from ALL spells is huge if your dm ever bothers to throw any at you. This is a feature that halves the damage from the most dangerous enemy type in the game, and combined with the fact you also have Aura of Protection to give your Cha bonus to all your saves and to those around you, wyou are the ultimate any mage.

Variant Human is pretty build neutral though, mounted combatant is nice with the fact you can use Find Steed to always have a mount. How you spend your ASI is up to you, most people recommend polearm master as a way to get a bonus action attack, but honestly just boosting your stats might be a better choice. Build pathing though it's tempting to say delay a few paladin levels for a quick 1/2 level dip into warlock. Hexblade lets you ignore needing strength beyond speed penalties in heavy armour by letting you use charisma to make weapon attacks. Invocations can also be pretty nice so look through those to see what you like for utility or eldritch blast augmenting.

Paladin 1 -> Warlock 1/2 -> Paladin X if you plan to max out your charisma. Straight Paladin is viable otherwise since hitting level 5/6/7 is such a power boost and delaying those features hurts, so maybe dip after that point instead if you aren't in a rush.

I just finished playing an ancients paladins, and I really enjoyed it. Making my save against a fireball, then taking half damage and halving it again was delightful. And if you are mounted when you cast misty step, the mount comes with you, which was equally fun.

Personally, not a fan of mounted combatant in general, and for paladins in particular. Any time you go inside, and can’t be on your mount, really any time you’re not on your mount, you’ve got a feat which is doing nothing for you.
And even with all the boosts from it, mounts are pretty fragile. Auras and AoEs wreck them. But as a pally, you can just resummon it. You’ll only get what, 3-4 feats (if that) in the whole campaign, but you’ll get more spell slots every day. It’s much lower resource cost to summon it again than to try and keep it alive with a feat.

All make sense. I am thinking the Feat I'll take if not Mounted Combatant, would go with Tough or Resilient CON.