The way I've interpreted this is that the spell itself, the casting of it, takes the duration of one round, your current action. The effect itself does not take place until your next turn as stated in the spell text proper. Once your next turn begins the spell is active, so to speak, and for the duration of that round you gain the advantage listed.
Furthermore if concentration is lost between casting and actual use, you'd resolve it as you would normally for checking on a break in concentration.
Casting Time: 1 action
Range: 30 feet
Duration: Concentration, up to 1 round
You extend your hand and point a finger at a target in
range. Your magic grants you a brief insight into the
target's defenses. On your next turn, you gain advantage
on your first attack roll against the target, provided that
this spell hasn't ended.
So cast spell, wait for next turn, spell effect is active for that turn, profit.
Furthermore I'd treat an attack on that turn that a PC might be able to make as a bonus action as the "next turn" attack with advantage, aka the first attack roll against the target mentioned in the spell proper. But the text is pretty clear that it starts working on your next turn, so that would likely be up to the DM.