A player can't hang someone from a tree branch over multiple turns using Thorn Whip. The answer by MikeQ clearly explains why not in the 1st and 3rd steps of casting the spell on a target.
But there's another way this could have gone.
I have to assume that you are talking about an attempt by the player to perform a lethal hanging, since the player was trying to have Thorn Whip wrap around the NPC's neck.
Hanging is a method of execution, designed to cause death very efficiently via sudden tension on the rope around the victim's neck, causing the neck to snap, which breaks the spine, and often severs or catastrophically damages the spinal cord. In real life, this was managed historically by having some sort of platform (wooden gallows, a bucket, a chair, a horse, etc.) for the victim to stand on while the noose was tightened around the neck, and then the platform would be removed.
The standard method is to have the victim fall (or more simply, move away from the fixed end of the rope) at least the length of their height in order to maximize the likelihood of an immediate death. Otherwise, the victim may survive and asphyxiate over a span of up to several minutes.
As MikeQ's answer also states, you're the DM here, so you have ultimate say in when to bend the rules toward the direction of fun. The 2nd step in MikeQ's description of events mentions correctly that the spell does not specify how Thorn Whip wraps around a target to pull them toward the player. This is where creativity and 'rule of cool' comes into play in a homebrew game to trump RAW:
If the player wants to target an NPC's neck with Thorn Whip, you can let that happen. This is the very common kind of scenario where the DM should allow the player to roll to see if they are able to accomplish something unique or creative. How you run the check is up to you:
- you could require the player to hit two separate DCs: one to hit the NPC (or for the NPC to meet the spell save), and a higher one to see if they can aim the Thorn Whip well enough to have it wrap around the target's neck.
- you could have a passive spellcasting skill check -- if the player's Wisdom or Intelligence or Charisma is high enough, they're able to target the spell that quickly.
- you could simply rule that, due to the element of surprise or the fact that the player is hidden, they are able to cast the spell accurately enough to do this.
The key factor is that the player doesn't need to hang the NPC for multiple turns. That's not the point of hanging someone.
This is the bit that the other answers, thus far, haven't touched on:
If the player is able to wrap around the NPC's neck and pull him up (toward the player) in an instantaneous duration (which at the very least we can say doesn't take longer than 6 seconds, e.g. 1 turn), then the vine created by Thorn Whip would almost certainly cause a classic Hangman's fracture, which would cause the NPC to immediately go Unconscious, and their airway would be blocked by both this fracture and by the squeezing of the vine around the Carotid arteries. Additionally, as mentioned above, this often causes paralysis due to damage to the spinal cord.
Now, after the spell ends, the target would drop 10 feet, take 1d6 fall damage, and (most likely) no longer be Unconscious. However, the following things would be true at this point:
- they'd take 2d6 damage:
- 1d6 for the Cantrip (or more if you're a high enough level)
- 1d6 for the fall of 10 feet
- they'd be on the ground, prone, and would not be holding anything in their hands due to Unconscious conditions (including weapons, shields, etc.)
- their neck would be broken
- depending on how you rule it based on the above information, the NPC would also be also Paralyzed
At this point, the NPC might as well be dead. Even if he somehow survived, he'd certainly be no immediate threat to the party.
All of this only requires one turn to happen instead of several. This point has an added benefit when compared to your description of the player's attempted scenario:
In a game where I am DMing, if the player is hanging by a vine or rope for multiple turns in order to try and choke out an NPC at the other end of that vine/rope, I would rule that the player is 'voluntarily' Restrained; they can't move, and they can't use their hands for anything other than hanging onto the vine/rope.
This makes the dangling player an ideal target for enemies to attack if any of them can see and/or reach the player, as attack rolls against a Restrained target have advantage.
It's not really how the condition is supposed to work, but I'd probably offer the player an option upon being attacked to use their Reaction to try and dodge an incoming attack at the cost of letting go of the vine/rope. And in that event, if the player is at least 10 feet up, they'd still have to roll for fall damage, too.