Preheat oven to 350 degrees F (175 degrees C). Grease an 8-inch square baking dish.

Combine brownie mix, vegetable oil, eggs, and water in a bowl until batter is smooth; fold in chocolate chips. Spread batter into the prepared baking dish.

Bake in the preheated oven until the top is dry and the edges have started to pull away from the sides of the pan, 45 to 50 minutes. Cool brownies completely. Cut into bite-size pieces.

Beat cream cheese in a large bowl using an electric mixer until smooth. Beat 1/2 cup half-and-half into cream cheese until incorporated. Beat remaining 1 1/2 cups half-and-half and pudding mix into cream cheese mixture on low speed until pudding is well blended, about 2 minutes.

Beat cream in a separate bowl using an electric mixer until soft peaks form; fold into pudding mixture.

Place a layer of brownie bites in the bottom of a trifle bowl or a large bowl with straight sides; top with a layer of pudding mixture and a layer of chocolate-covered nougat bites. Continue layering 2 or 3 more times in the same order with remaining ingredients.