Possibly, it depends on the trigger condition and on what perceivable circumstance are.
If your readied action is set on a trigger that happens before or after the turn of the enemy, then only one turn (the next turn that has not happened) is affected. If your trigger is set in a way that it triggers during their turn that (current) turn and the next turn is affected. Your emphasis is a correct plain-English reading.
Now we got a little problem because of how Ready (PHB 193) is written:
you decide what perceivable circumstance will trigger your reaction.
When the trigger occurs, you can either take your reaction right after
the trigger finishes or ignore the trigger. Remember that you can take
only one reaction per round.
Can your gunslinger anticipate the attack before the enemy attacks?
This is an issue that has been debated here, here, and here and at other places, the issue heavily hinges on what perceivable circumstance are.
There is no clear consensus, make sure to discuss what perceivable circumstance are with your group before the issues arise at your table and see how it affects scenario three and scenario four.
I have taken the liberty to address scenarios that do not directly deal with your issue to make the wording and working of trigger conditions more apparent.
Remember that you have to declare your Trick Shot before the attack roll.
Trick Shots. You learn two trick shots of your choice, which are
detailed under “Trick Shots” below. Many maneuvers enhance an attack
in some way. Each use of a trick shot must be declared before the
attack roll is made.
Scenario one:
You declare to ready your action to shoot the cultist on sight.
The cultist loses their invisibility due to some reason on a turn that is not the cultist's turn and before your next turn.
The outcome: you shoot the cultist right after they become visible and you declare your Trick Shot before the attack roll. The cultist is affected for exactly one of their turns, which is the next turn (and until the end of that next turn).
Scenario two:
You declare to ready your action to shoot the cultist on sight.
It is the cultists turn before your next turn and the cultist moves from behind their cover into sight.
The outcome: before the cultist can spend their next action but right after the trigger finishes, you shot the cultist and declare your Trick Shot before the attack roll. The cultist is affected for the rest of their current turn and until the end of their next turn.
Scenario three:
You declare to ready your action to shoot the cultist when the cultist attacks you.
It is the cultist's turn before your next turn, and the cultist finishes their turn by attacking you with their scimitar.
The outcome: the cultist spends the last action of their turn to attack you. Right after their attack (right after the trigger finishes) you shot the cultist and declare your Trick Shot before the attack roll. The cultist's turn ends before they are affected by your Trick Shot. The cultist is affected for exactly one of their turns, which is the next turn (and until the end of that next turn).
Scenario four:
You declare to ready your action to shoot the cultist when the cultist aims their scimitar at you.
It is the cultists turn before your next turn, and the cultist finishes their turn by attacking you with their scimitar.
The outcome: the cultist spends the last action of their turn to attack you. Before they can attack you, right after they aim their attack at you (right after the trigger finishes) you shot the cultist and declare your Trick Shot before the attack roll. The cultist is affected by your *Trick Shot for their current turn, before their attack with the scimitar and until the end of their next turn.
For the turn order, read The Order of Combat (PHB 189-190) for reactions read Reactions (PHB 190).
