Preheat oven to 425 degrees F (220 degrees C). Spread potato nuggets on a baking sheet in a single layer.
Bake in the preheated oven until browned and crisp, 20 to 25 minutes. Transfer to a large bowl; season lightly with salt.
Mix pulled pork and barbeque sauce together in a bowl until pork is evenly coated.
Whisk sour cream and smoked paprika together in a small bowl.
Melt butter in a saucepan over medium heat. Whisk in flour until a paste forms, 2 to 3 minutes. Whisk in milk until no lumps remain. Cook and stir until sauce thickens enough to coat the back of a spoon, about 5 minutes. Remove from heat; whisk in Cheddar cheese until melted. Season sauce with salt and pepper.
Drizzle Cheddar cheese sauce over potato nuggets. Cover with pulled pork mixture and dollops of sour cream. Top with pinto beans, roasted peppers, green onion, and thyme.