Preheat the oven to 375°F (190°C).
In a large skillet over medium-high heat, cook the ground beef, chopped onion, and chopped green bell pepper until the beef is browned and the vegetables are tender, about 6-8 minutes. Drain excess fat.
Add ketchup, water, tomato paste, Worcestershire sauce, brown sugar, yellow mustard, garlic powder, onion powder, black pepper, and salt to the skillet. Stir well to combine.
Reduce heat to medium-low and simmer the sloppy joe mixture for 5 minutes, stirring occasionally.
Spread the sloppy joe mixture evenly in the bottom of a 9×13-inch baking dish.
Sprinkle 1 cup of shredded cheddar cheese evenly over the sloppy joe layer.
Arrange the frozen tater tots in a single layer on top of the cheese.
Bake uncovered in the preheated oven for 30 minutes, or until the tater tots are golden brown and crispy and the cheese is melted.
Remove from oven and sprinkle the remaining 1 cup of shredded cheddar cheese over the tater tots.
Return the casserole to the oven and bake for an additional 5 minutes, or until the cheese on top is melted.
Remove from oven and let the casserole rest for 5 minutes before serving.