Pulse vanilla wafer cookies in a food processor in batches until finely ground.
Combine cream cheese, confectioners' sugar, butter, and vanilla extract in a large bowl; beat with an electric mixer on low speed until moistened. Increase speed to medium and beat until smooth. Add milk and vanilla pudding mix; beat on low speed until smooth. Fold in whipped topping with a spatula.
Layer wafer cookie crumbs and pudding mixture in a large bowl.