This may be unnecessary
Hopefully between the other answers, and some of the links I suggested, you have a better understanding of surprise, initiative, and how they interact.
But let's return to your original question - your ambusher has lost initiative; their target is still surprised, but now has a reaction available. Should you cease your attack? Has it been hopelessly compromised, so that your surprise is less effective? Or can you go ahead with equal chance of success?
The answer depends on the reactions available to them
Certain special abilities, spells, and situations allow you to take a special action called a reaction. A reaction is an instant response to a trigger of some kind, which can occur on your turn or on someone else's. The opportunity attack, described later in this section, is the most common type of reaction.
Note that you may only take a reaction when something allows you to. Each target will have access to different reactions, depending on their "special abilities, spells, and situations". Most targets will have access to an opportunity attack as a reaction - will this hinder your ambush? Well, it will have no affect on your attack itself - but it might affect what happens next. If your ambush was with a ranged attack, the fact that the target can use its reaction on an opportunity attack is irrelevant. But if you needed to close to melee, this could be important. If your attack did not incapacitate them, and they have a reaction available, they certainly could take an opportunity attack on you if you moved away.
Another common type of reaction is one that is used to take a Readied Action. This, however, you do not have to worry about at all. Because you surprised the target, they had no Actions on their first turn. Thus they were unable to Ready an Action and even though they have a reaction available on your turn, they will have no readied actions to use it for.
Beyond these two general types, the reactions available will depend on the specific target and what they have access to - three that you suggested were the Shield spell, a bard's Cutting words class feature, and a Battlemaster's Parry maneuver. Interestingly, some of these might make you call off your attack and others not.
Crucially, you have posited that your attacker starts their turn invisible. Now look at the description of Cutting Words:
Also at 3rd level, you learn how to use your wit to distract, confuse, and otherwise sap the confidence and competence of others. When a creature that you can see within 60 feet of you makes an attack roll, an ability check, or a damage roll, you can use your reaction to expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and subtracting the number rolled from the creature’s roll.
Notice that the bard can use this only on attackers that it can see. If you start the combat invisible, even if you are using the lowly second-level spell, you will be invisible until your initial attack hits or misses. Thus if your target is a bard, even if they have their reaction available, they will not be able to use their cutting words on your first attack, because they will not be able to see you. If you were planning on taking them out with a single attack, you have lost nothing by losing initiative. If you were counting on a second attack on your turn, such as with Extra Attack or a bonus action, that would be a different calculus. In that case, beating their initiative would confer an advantage, since they would not have a reaction available to use cutting words even though they could see your second attack.
Now compare this situation with the language of Shield, where the target can take their reaction:
when [they] are hit by an attack or targeted by the magic missile spell
and the parry maneuver, where:
When another creature damages [them] with a melee attack, [they] can use [their] reaction and expend one superiority die to reduce the damage by the number [they] roll on [their] superiority die + [their] Dexterity modifier.
Notice that these reactions are triggered by being hit (Shield) or damaged (Parry). In neither case does the target need to see you. Not all reactions are equal, and losing Initiative would be worse against some targets compared to others.
However, lest you think that this depends on your ambusher being invisible, be aware that the same consideration may be true if they are simply unseen. Refer to the Hiding sidebar in PHB Chapter 7:
In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you. However, under certain circumstances, the DM might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack roll before you are seen.
Depending on the circumstances of your ambush, even if you are not invisible and have lost initiative, you still might be able to approach your target unseen. Suppose you are approaching the target in a dense forest. You have a good Stealth roll; the DM determines that they are distracted (perhaps singing to themselves, composing their next ballad?) and surprised. You move at them from behind and enter combat; the DM instructs you to roll Initiative, and you lose. A bird sees your movement and takes to the wing; your target startles and now has their reaction - but they still don't see you moving up behind them until after your first attack. Bards beware!