Looking at the rules for the 5e Assassin subclass (and similarly for the PC Bugbear Surprise Attack feature),
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.
and similarly for the Bugbear race from Mordenkainen Presents,
If you hit a creature with an attack roll, the creature takes an extra 2d6 damage if it hasn’t taken a turn yet in the current combat.
I wasn't sure whether an attack made as a reaction during the creature's turn would still receive advantage from this feature. My instinct is that a turn is considered "taken" at its end, and this part from the PHB
The Order of Combat
A typical combat encounter is a clash between two sides, a flurry of weapon swings, feints, parries, footwork, and spellcasting. The game organizes the chaos of combat into a cycle of rounds and turns. A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn. The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative. Once everyone has taken a turn, the fight continues to the next round if neither side has defeated the other.
seems to indicate that this is the case, since that last character's turn has to be completed in order for combat to continue after it has taken its turn.
The hypothetical I'm thinking of is a Bugbear Assassin rogue with 2 levels in fighter that gets the drop on the enemy, surprising the creature, and the rogue goes first, with the creature taking its turn immediately after the rogue. The rogue attacks and gets all the extra damage features etc., then uses Action Surge to take the Ready action.
The rogue readies an Attack action with the trigger being the start (or middle) of the creature's turn, with the intention of benefiting from Sneak Attack a second time this round by attacking with a Reaction after the rogue's turn has ended. The creature should still be under the effect of surprise (as it cannot take actions until the end of that turn), but would the Surprise Attack feat and advantage part of the Assassinate feat still be active after the creature's turn has started (but not ended)?
I imagine the trigger could be rephrased to, "just before the creature takes its turn," to resolve ambiguity, but it seems like we run into trouble in getting that second Sneak Attack damage if the creature goes immediately after the rogue–this is in contrast to the rogue taking their turn, some third creature taking its turn, and then the creature in question taking its turn. As near as I can tell, there is no "in between" time in combat from one turn to the next, meaning in this situation either the creature has started its turn or the rogue has not ended their turn.
So, in the situation of rogue then creature goes, would this readied Attack benefit from the advantage in Assassinate and the bugbear's Surprise attack damage? Again, the common sense answer feels like yes, but I want to know if that's how it would be adjudicated vis-à-vis RAW.