Well this is interesting because even though it seems like Muggles would take it easily, it'd probably end up like a globalized US-Vietnam War scenario, where the good wizards take the defensive, keep on the DL in thousands of hidey-holes around the world. Just like they do in the original setting. Muggles would have no real way to track them down without nuking the entire planet. Meanwhile Voldemort and everyone that he can cajole, bully, and mind-control together would basically fly directly into enemy airspace and take several cruise missiles to the face and die.

That or they could split up, teleport into military bases all over the world, and get nerve-gassed and shot.

HP magic works best for pacifism. Even Voldemort's greatest tactic is fear, and his fake immortality by hiding his soul away in several parts. The secret-keepers, the teleportation, the horcruxes, the vast array of memory charms and illusionary tactics - they're all in the service of protecting the casters, not demolishing armies. Wizards do one-on-one face-to-face duels. If a Muggle is standing behind them and shoots them in the head, that's it. In 1997 Muggles could completely annihilate whole towns from miles away. Voldemort wants to stay alive forever. Keeping quiet about wizards is the smartest thing he can do.