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.
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.
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.