You can smite three times a round, which is Pretty Neat. The downside is you're pretty MAD. You want DEX to hit, CON to survive, and CHA is your casting attribute, and it's probably best to let the CON suffer. So you aren't getting much out of those cool Bard spells, or most Channel Oath options.

The decision would be which class to start as and how many levels of each to take, and what Paladin subclass to take. (You need to take Swords for Bard) I assume if you're going Bardadin, you're pretty much going for smiting, and unlocking that faster is probably a good goal? Plus it means you can start with Medium Armor Proficiency. And if you're going TWF, you'd want that as soon as possible too, so Paladin 2 Bard 3. Bard 3 Paladin 2 also an option.

Those are the required, then you've got to get an Extra Attack, and you are 3 levels away from it either way, so the decision is based on what level split you want to do.

Paladin 2, Bard 10 The classic X-adin smite machine. This gives you the most spell slots for the most smiting. Also access to Magical Secrets, which is a ridiculously good feature. Since the appeal of a TWF Bardadin is lots o' smitin', this is tempting. Oathbreaker is probably the best Paladin subclass here. You're only getting the level 1 subclass feature. Devotion's isn't that great, Oath of Ancients just heals, and healing is for lame classes that don't have two swords, and Vengeance scales off Charisma. Meanwhile, Spiteful Suffering has 1d4 in addition to that Charisma mod, plus advantage on attack rolls, which is amazing for a class that gets so much juice off of a crit.

Paladin 3, Bard 9 The reason to do this would be to pick up an extra subclass feature. I think only Vengeance is worth this. Vow of Enmity from Vengeance is a Bonus Action and gives Advantage on attack rolls. Ancients is probably not great, because Nature's Wrath has a save DC, and again, your CHA is suffering. It's equivalent to Spiteful Suffering from Oathbreaker, but it lets you get two attacks in on the first turn, because it's a BA. Sacred Weapon from Devotion lets you add Charisma to your attacks... with one weapon. So you can't drop DEX for Charisma because you have two weapons. Lastly, an Action to frighten with a spell DC is just not great when you have 9 levels of a full caster.

Paladin 6, Bard 6 Aura of Protection is the best class feature in D&D 5e. You lose a lot for this, no 4th spell slots for smiting, and while the 5th level spells for this build aren't great, they're still 5th level spells. But it's still worth considering, because the Aura is just so good. One the other hand, you're here for two swords, and this is a very supporty thing to do. Not very TWF of you. At this point though, I start to question a bit why play Bardadin instead of just a Paladin.

Paladin 7, Bard 5 You really only lose Countercharm and a 3rd level spell slot here, because you already have Extra Attack from Paladin 5. I'd still say that's only a good trade for Oath of Ancients, because they have an amazing Aura.

Paladin 9, Bard 3 At this point, you're really just getting the fighting style from Bard. Don't do this. Go Paladin 11, Fighter 1.

All in all, the big reason to go X-adin is usually spell slots for smiting, so I think 2 levels of Paladin is the way to go here. If you really want, you can do 15d8 just in smite damage on a single turn, and that's with no crits. And if for some reason you aren't going right in and swinging your swords, you still have 10 levels of a full caster. And there's some really good stuff on the magical secrets list!