How Does Assassinate Work when Reinforcing a Fight that has already started?

Question

The Rogue Assassin sub-class has the Assassinate ability as its first subclass feature at 3rd Level, with text that reads:

Starting at 3rd level, you are at your deadliest when you get the drop on your enemies. You have advantage on attack rolls against any creature that hasn’t taken a turn in the combat yet. In addition, any hit you score against a creature that is surprised is a critical hit.

My question with this is what happens when part of the Party is already in a fight and the Rogue arrives and begins to fight?

Let's say the Rogue got split from the rest of the Party. The rest of the Party gets spotted and engaged by a group of sentries and begin combat, but the Rogue isn't in the combat area. Two turns into Combat, the Rogue finishes whatever he/she was doing and then moves up and goes to enter combat by taking a shot at one of the sentries.

How would Initiative be rolled in this scenario, and how would the Assassinate feature be determined?

Does it count creatures who had not yet taken a turn in combat, or creatures who had not yet taken a turn in combat with the Rogue?

Answer 1

A rogue entering a fight late will only be missing out on potential assassination targets

How to deal with combatants entering a fight after it has already started

This part of your question is already addressed in another Q&A: How to handle initiative when a new force joins a combat in progress.

The answer is simply that when a new combatant enters the battle, they roll initiative like normal and go on that initiative.

This is also the same suggestion that Jeremy Crawford has:

Honestly, we never should have had spells tell you to roll initiative for creatures. The general rules just need a line that says, "If a creature joins a battle that's already in progress, roll initiative for the creature as normal."

When a combatant enters a battle late, they do not change who has already taken a turn or acted in the battle before their arrival.

Assassination effects

There are two effects that you can get from assassinate: advantage against enemies who act after you and automatic critical hits against surprised enemies. For different reasons, both effects only care about when the battle starts and not when the rogue enters it.

Advantage

You have advantage on attack rolls against any creature that hasn’t taken a turn in the combat yet.

This does not not say "taken a turn in the combat against the rogue". In fact, that statement doesn't even make any sense. You either have taken a turn in combat or you have not. There is nothing that measures how many turns you have taken in relation to a specific creature.

It is pretty clear that this ability is supposed to be used at the beginning of a fight, not at the beginning of the fight with respect to the rogue.

Jeremy Crawford has agreed with this:

Assassinate works only during the first round of a combat, unless a creature joins a fight later and has a lower initiative.

Note that by "a creature" he is referring to a potential target of assassinate. If an enemy joins the battle late, they will be having their first turn in combat and assassinate will work on them. If the rogue enters a battle after everyone has already taken their turns, then they do not get to assassinate anybody already in it.

Automatic critical hits

In addition, any hit you score against a creature that is surprised is a critical hit.

The rule on suprise says:

Any character or monster that doesn't notice a threat is surprised at the start of the encounter.

After the battle starts, there are no more surprised enemies. So, joining late also means no automatic critical hits.

Thus, a rogue entering a fight late will only be missing out on potential assassination targets.

Answer 2

Initiative for the Rogue would be rolled on the round when he joins the fight. Sorry, but surprise is only relevant before the fight begins, when some of the combatants-to-be fail to notice the enemy before the fighting starts. Likewise, everyone involved in the fight has already taken a turn, so that automatic advantage doesn't apply either. The best the Rogue can do is to sneak up and attack with advantage from hidden, and add Sneak Attack damage for that, or attack an opponent who is already engaged by a party member, and get Sneak Attack damage that way.

If there are other sentries at nearby posts, say more than short bowshot range and not firing into the battle, and who have not left their posts, not rolled initiative, not actually joined the battle, then I might let the Rogue have the opportunity to "get the drop on" those, if he plans and approaches carefully.