Heat oven to 400°F. Lightly grease a 9x13-inch baking dish.
Warm olive oil in a large skillet over medium-high heat.
Brown the ground beef, breaking it up, until no longer pink, 5–7 minutes. Drain excess fat.
Stir in onion and garlic. Cook until softened and fragrant, about 3 minutes.
Season with Worcestershire, paprika, salt, and pepper. Cook 30 seconds, stirring.
Whisk soup, sour cream, and milk in a bowl, then pour into the skillet and stir until creamy.
Fold in 1.5 cups cheddar. Spread the beef mixture in the prepared dish.
Arrange tater tots in a snug, even layer on top.
Bake until the tots are deep golden and the edges bubble, 30–35 minutes.
Scatter remaining cheddar over the tots. Bake 5 minutes to melt. Rest 5 minutes, then garnish with green onions.