Watching the release trailer again, the paladin's smite (timestamp 0:56) seems to take two seperate animations to complete, much like other spells. My guess is smite will work like battle master maneuvers, with spell level selection pre-attack and no spell slot spent on a miss.
I'm not sure what that says about reactions, but unless they change the paladin wouldn't be able to smite on an attack of opportunity.
Could also be a smite spell, not divine smite. At the end of the scene, it seems like the enemy is left glowing, which would fit with branding smite
