For example, in One Piece, Brook, and sometimes Zorro, do some big attack with their swords, but the cuts are not shown until after they return their swords to the sheaths.
This trope is usually attributed to Fist of The North Star, after Kenshiro attacks someone he pauses and says "You are already dead". It is only then that they realise their defeat and the attacks are applied.
It's similar to how in Road Runner, Wilee Coyotee often runs off of cliffs, without realising the ground below him has stopped - he only falls when this is pointed out to him.
The usual rationale is that everything happens so fast that the character cannot register what is going on - and neither does their body.
This slashing technique is explained in Chapter 450: General Zombie Night. known as Hanauta Sancho: Yahazugiri (literal meaning: Three verse humming: Arrow-notch slash). The slash occurs at such a fast pace that it appears as if the swordsman simply walked past the victim. The victim feels the effects only after a certain amount of time, which is about the time to walk three metres (or ten feet) away. As Dr. Hogback points out, the victim continues to hum three verses, and only then realize that they have been cut. Further, he notes that only a "master" swordsman can use this slashing technique.
The swordsman sheathing the sword and announcing "I have already cut you", moments before the victim feels the slash is only because it looks cool. The victim would feel the effect all the same even if the swordsman just stood there doing nothing after slashing.
TV Tropes defines this as Delayed Causality. It is purposely meant for dramatic effect.
The in-universe explanations for it (if any) are often noticeably tongue-in-cheek rather than meaningful. There is nothing believable about cutting someone up with a sword and them failing to notice.
In One Piece this is more strongly associated with Brook, because his fighting style has emphasis on speed and "so fast they didn't notice" plays to it.
Did you ever experience being cut or injured and you notice it later because the cut or injury was insignificant that you fail to notice it until it's pointed out to you or you feel pain later due to a movement that depends on the place where you get injured or cut?
It is probably inspired by these kind of situations in real life but with a little exaggeration (being that the cut is critical that you can't possibly be that stupid that you can't notice and you can pretty much predict the exact moment that the victim will feel the injury every time without fail)