Yes, but the knowledge of the trigger can be given to players, not characters
As you correctly point out, RAW a character does not know that they have been targeted by a scrying spell. But if that character has an ability that allows them to react to just such a targeting, or allows them to react to something that forces a saving throw they don't know about, there is just the difficulty or contradiction you describe.
As a DM in this situation, you have a few choices.
1. You can not describe the triggering event and thus keep the character from using their ability. While this does adhere to rules as written, you are basically deciding a rules contradiction in favor of the NPC and against the character. Of course, it goes both ways, and the characters can do this in their own scrying. But given that the features you mention are class-specific, and the players likely chose their class with certain features in mind, it would not be surprising if some of the players feel that you are unfairly 'nerfing their character', since it will limit them every time and limit the NPC's only in the rare case that you have made them a class with one of these abilities.
2. You can assume that the character feels something on the subconscious level, and decides to use their ability without being aware of it - no character knowledge of the trigger is needed. While this at least allows the character to use their ability, it robs the player of the agency in making the decision.
At 15th level, you've learned how to magically chastise anyone who dares cast unwanted spells at you and your wards. Whenever you or a creature you can see within 30 feet of you succeeds on a saving throw against a spell, you can use your reaction to deal 2d8 + your Charisma modifier force damage to the spellcaster.
Since there is no per day / per rest limit to the use of this ability (it may be used 'whenever') it seems at first glance that one way out of requiring character knowledge is just to say 'Your paladin does this instinctively. It is a reaction, after all, and most reactions happen faster than thought. We can assume that any time something forces a save on you or your allies, you use this ability if you can, even if you don't realize that you used it.' That may often work narratively to spare you having the character actually know about the saving throw. But even if you do this, each PC gets only one reaction per round. Action economy means that sometimes their would be a choice between using this ability or taking another reaction. Do they punish a minion for casting a spell on an ally, or take an opportunity attack on the Big Bad?
When you or another creature you can see within 30 feet of you makes an ability check or a saving throw, you can use your reaction to add your Intelligence modifier to the roll.
You can use this feature a number of times equal to your Intelligence modifier (minimum of once). You regain all expended uses when you finish a long rest.
Here the stakes are even higher. Not only does the player have to consider the opportunity cost of making a reaction within the action economy, but they have to decide whether or not to spend a resource that is limited to a few uses per long rest.
3. You can tell the player as much as you like about what is happening, and expect them to compartmentalize that knowledge from the character. It might be as explicit as "Okay, someone is trying a divination spell on one of you that gets a save - does the paladin or artificer want to do anything about it?" and then you trust that they don't start trying to actively find magic that blocks divination. Or it could be as subtle as "Your subconscious mind detects the presence of malign forces of which you are not consciously aware - do you have an intuitive response to protect yourself and your allies?"
You know your table and you know how you want to run your game, but obviously I think option #3 is the way to go. Removing that decision from the player because their character would not realize a save was being made says you can't trust your players with metagame knowledge. Telling the player 'I'm going to give you just enough information so that you can fairly make your decision, but in return you have to have your character act as if they don't know what save was made within the boundaries of allowable metagaming at our table' says that you trust the players to keep their end of the social contract. I believe the DM should provide the player with just enough information to make reasonable decisions, even if that player then has to pretend that their character didn't know exactly what is happening. Not allowing players access to the information they need to use their abilities robs them of agency. Ultimately it is the players who are making the decisions, not their characters.
Decisions are made by players, not characters
Players can try to think in terms of 'how would my character react / decide based on just what they know', and this can add depth to their role-playing, but ultimately it is still the players making the decisions, and some degree of metagaming is inevitable. Different tables have different styles of play, from min-maxers playing a strategy game with the Monster Manual open at the table, to story-tellers where the DM rolls all the dice, including for players, in secret (or rolls are made by players in advance and handed to the DM at the start of a session) so that players don't know how well they did beyond the narrative. The amount of 'pretending to not know' varies between groups, but even at tables that greatly restrict player knowledge, some metagaming is inevitable. This question has some interesting discussion of a kind of reactionary metagaming, where the players are trying so hard to show that their characters don't know some things that their behaviors become nonsensical within the context of the game world.
There is lots of in-game assistance for those who enjoy a style of play that focuses on in-character knowledge and motivation. For example, spells and abilities often are described with in-character 'fluff' first and game 'crunch' after. Consider the description of the bard's ability to use Cutting Words:
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.
The second part is what the player knows, the first part is what the player pretends the character knows. It is not what the character actually knows, though, because the character doesn't actually know anything. The character is a fiction. The player pretends the character knows or does not know things in order to achieve a certain style of play.
Consider the OP question's statement that "Characters don't even know what saving throws are." Now imagine your wizard deciding in-game whether or not to learn "Mind Sliver" without admitting character knowledge of saving throws:
You drive a disorienting spike of psychic energy into the mind of one creature you can see within range. The target must succeed on an Intelligence saving throw or take 1d6 psychic damage and subtract 1d4 from the next saving throw it makes before the end of your next turn.
Some players are content to say "Spells that debuff saves are great, I'll take it!" Others may find themselves saying "The first part of the spell damages the psyche of my opponent and the second...makes them...unlucky...when they are challenged by certain...things they have to resist."
This same wizard who doesn't know that saving throws exist, nevertheless knows that two mind slivers won't make someone more 'unlucky', but a mind sliver and a bane together will make them 'worse at resisting things'. You can keep pretending they don't know what a saving throw is, but it does become burdensome if you need to talk in-character all the time.
But in neither case are we talking about character knowledge - we are talking about how a player wants to portray character knowledge. Find a style of description that is the most fun and fair for you and your players. Find a style that works at your table, and don't worry too much about whether RAW a character knows something.