For the secret sauce: Combine the ketchup, apple cider vinegar, Worcestershire sauce, garlic powder and hot sauce in a small microwave-safe bowl. Microwave until just simmering, about 30 seconds. Keep warm, covered.
For the sandwiches and Cheddar sauce: Lay the slider buns cut-side up on a rimmed baking sheet and bake until warmed through, about 8 minutes.
Meanwhile, combine the processed cheese, Cheddar, milk and paprika in a small saucepan and warm over medium-high heat, stirring, until melted and smooth, about 2 minutes. Take off the heat and keep warm, covered.
Spread both sides of the buns lightly with the secret sauce. Place a few slices of roast beef on each bottom half of the buns. Rewarm the cheese sauce if necessary, then spoon some of the cheese sauce on top of the meat. Cover with the top half of the buns and serve immediately.