I had an interesting idea for a roleplaying hook recently, but it brought some mechanical problems with it. I had the idea of an npc with a greatsword chasing after a smaller npc, say in a tavern. He attacks his target, and one of the PC's deflects the attack with their own weapon, knocking it aside.

How would you apply something like this? Would you allow the PC to intercept the attack because it allows them to feel powerful or cinematic, or would you apply standard rules, e.g "How does your character have the ability to do that?".

Turn order is just a simplification of combat so rules make sense. Perceivably, this could be a character using the ready action to move between the attack and the target and give the target half cover.

You have one npc attacking another npc. It's up to the players if they want to interfere or not. A DM shouldn't say that a player interferes or not.

So have the npc attack the other. If you want to persuade the players to get involved then have that greatsword attack be close enough to a player that the player gets a small piece of hair cut off. On a successful dex throw. Fail the dex throw and player also takes damage. 

Just that npc swung wildly and almost took a player's head off. Having the fight near to the players might get them involved if the fight may cause then harm for no reason.

Assuming the cinematic was the chase and one of the players unexpectedly tried to intervene with a deflection, I would say you can call for a skill or stat check... athletics maybe,  or dex? Assign advantage or disadvantage as appropriate.