They wouldn't care at first. A Psi Operative is unlikely to kick off the surveilance the Ministery has for magic being used.
It will take a good while before the news about some muggle using magic-esque abilities without a ward, and there's no doubt in my mind that even muggleborn adults will at first dismiss these psi-users as charlatarns. After all, how could these silly muggles ever hope to match them?
Once the realization kicks in, hell will break loose for a long while during which the various wizarding goverment will try to abduct the psi-users and mind-wipe everyone else because obviously muggles can't be trusted with these abilities. The Operatives will not comply with that bullshit, and six months later, you have a nasty war as aurors and psi operatives duke it out with the muggle and wizarding citizens being caught in the crossfire. Even if a cease fire is negotiated, the bitterness will remain.
The cultures of non-magics will shift from "Magic isn't real" to "Fuck these assholes!" while the wizarding cultures will go from "Silly animals, thinking they are people" to "Oh shit, they can and will fight back!", with muggleborn wizards being bullied and wizardborn psi-users being viewed with suspicion.
Wow...you are cynical.
And yet...some of that certainly rings true. Thoughts
@Rhyse?
Sorta? Most wizards outright don't care about muggles from what we see in the books. Voldemorts psychotic little brain trust though, yeah, they'd definitely organise shit like this. I doubt the actual ministry would though; from what we see of them, they're just apathetic to muggles at worst, and at best are outright friendly. With the Minister of Magic prioritizing the safety of the muggle PM over many of his own workers.
The problem is - as always - the other side has magic. In real life, the government has the absolute monopoly on violence. People can LARP otherwise, but the government can drop a JDAM on you, and there's literally nothing you can do about that. This isn't the case for wizards. The government passes laws, but wizards don't seem to live near each other, so any enforcement of those laws is weak as hell. You can't break up meetings when the meetings can happen in the mirror universe - new fantastic beasts is wild - and you can't stamp out rebellious movements when they can kill you as easily as you can kill them. In real life, the government having all the infrastructure, means of production and cities makes them hard targets for random civvies that want to fight them; for wizards it's the reverse. A wizard can fuck off to the woods and from there blow any and all targets he so chooses to hell with impunity. Any sort of 'mass action' by the wizard government seems limited to about a dozen people, given free reign to 'deal with' problems that arise. Aurors are under the control of the DMLE sure; but what that looks like in practice is wizards picking targets, teleport raiding them, and returning to base under their own initiative. Which doesn't make for solid top down control.
(This is of course, all inferences from the books, Rowling didn't say any of this; because this is super autistic lol. I'm trying to rationalize the little we see of their population structure)
I'd say it might cause a civil war among the wizards. A proper one as well, none of this coup bullshit that Voldemort pulled. When Voldemort did his thing, he kept it fairly quiet. I'm sure a lot of people knew the truth, but Voldemort wasn't in office. The illusion of democracy and normalcy remained. If the minister ordered the abduction, or attack of random muggles who appear magical - in the same way Hags, Giants, Vampires, Merfolk, and others are magical - people would shit a brick. Voldemort initiated a limited pogrom while he had peoples children hostage, while his men had infiltrated government and were mind controlling people, while he was the single most powerful wizard around and had come back from literally dying. Half the population still went against him, and his cadre of diehard consisted of about fifteen people - most of whom were mentally ill - that stayed loyal when things got tough.
That was a man so powerful that his temper tantrum caused an Earthquake while he dueling Flitwick. To fight Voldemort was to just die, outright. You lose, do not pass go, straight to the grave. If you beat him, congrats, cunts immortal he'll be back, if you lost, congrats, your family is now dead. It was, to the average wizard, a lost cause. Even when it was easy as hell to surrender, loads of them still fought with the intention of dying to spite his cause.
Fudge, and Umbridge, and people like the Carrows aren't Voldemort. The second they started actively rounding people up, the country would shit the bed and there'd be hell. None of that 'Oh we'll go along and keep our heads down'. It'd be 'Hang the fuck on, I'm not taking orders from this prick! I'm a better wizard than he is!'. The more I think about it, the more convinced I am, that not only is the statute of secrecy done for; but that the government of wizarding Britain might also be done for. Compelling people to work at gunpoint already barely works, it's not going to work at all if they can teleport.
That then of course opens up a lot of issues since the muggle world is going to soon learn that the assumption of 'Humans are the only intelligent life on Earth' is hilariously wrong. Of the top of my head: Hags, Trolls, Vampires, Giants, Veela, Sphinxes, Acromantula, Centaurs, Merpeople, House-elves and Goblins are all at least as intelligent as a human is. On top of that, the economy is going to go tits up. There's an entirely alien banking system that exists alongside the normal ones. Wizards make most resource jobs almost entirely obsolete. Medicine would be revolutonised, then made horrific when carefully controlled outbreaks of magical sickness become less controlled as thinks break down - not even sure how to diagnose 'bits of the patient keep teleporting away randomly - which would not be helpful.
Muggles can't really control anything as shit spirals. They can't be bombing their own cities, Psis being able to see through anti-muggle charms doesn't help when some Voldermort Stan teleports into Piccadily circus, shouts 'Merlina Ackbar!!' and blasts apart fifty people, then teleports away. If we're talking world wide, there'd be reactionary movements springing up everywhere. People supporting Voldemorts ways, people supporting Grindelwalds ways, people demanding the end of the statute formally, people not caring and going full 'Sovereign citizen' and breaking it openly. Even if Dumbledore is still alive, I'm not sure what he'd do honestly. The man seems to be single goal oriented around 'Kill Lich'. I could see a lot of muggleborns going to the news, or the government and outright saying 'wanna hire a wizard?'. Actually, in general I could see that happening, pottermore notes that fucking Rolls Royce have a magical department because they sell enchanted cars. (Or the pureblood brought a car and enchanted it, either way shows that 'ooh, shiny toy!' is an effective way to bribe wizards.).
The wizarding world as Rowling presents is gone though. There's no coming back for it, the only question is if it goes slow or fast.
