They can take reactions. The rules don't state surprise has worn off. That's the issue of debate.
That's why I said the effect of surprise, namely the lack of being able to take a reaction, has worn off.
Yes, there is debate as to whether
surprise as affects the assassin ability
Assassinate has worn off at that point. Personally I think it has worn off for the reasons I've already stated.
First, surprise is very concisely explained in the PHB:
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.
Just considering the general case of surprise and imagining a game with no assassins, it doesn't really make sense to talk about being surprised outside that range of time because there are no other effects whatsoever.
Secondly, adding consideration for the assassin class ability Assassinate:
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.
The debate comes from the last sentence: Is a surprised target who's turn has passed (and can therefore take reactions) still surprised?
Some say no, some say yes.
No: A fast assassin gets advantage and an automatic-critical. A slow assassin gets a normal attack.
Yes: A fast assassin gets advantage and an automatic-critical. A slow assassin gets no advantge yet still gets an automatic critical.
Regardless, you still have to track initiative during the surprise round to determine whether or not the assassin gets advantage.
The end result is hardly earth shatteringly different played either way, but I think 'no' gives a cleaner more cohesive result.
I think the second part of that ability is meant to apply "in addition" to the first, just as the sentence says, i.e. it doesn't occur when the assassin is attacking something which has already had a turn in combat. It also seems backward to think that adding the Assassinate ability to the game would extend the noticable length of time a target is surprised compared to the surprise rules alone. The possibility that a slow assassin would not get advantage and yet still get an auto-critical seems weird and slightly clunky to me too.
Somewhere someone posted that a (possibly) official clarification ruled it was intended to work as I suggest, although honestly I don't care so much about that, only wanting an internally consistent game world that is easy to run.
