PHB, pg 189 - If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends. That's it. Initiative is rolled as normal for the round but surprised creatures can't act or react, and can also be targets of assassinate. Nothing changes about the structure of the combat round or the order of operations. To break that down a little, the consequence of being surprised is that you don't get an action or reaction in that turn. So the first turn of the combat the creature is surprised, but in the second round they are not, regardless of whether or not they've taken an action yet. The wording of the part about reactions makes it pretty clear that the duration is that first turn, because as soon as the second turn of combat starts they would be able to take a reaction and thus are no longer surprised.
Initiative does not, btw have any effect on the criticals doled out by assassinate, which apply in entirety of the first round.