Pulse chocolate sandwich cookies into fine crumbs in a food processor.
Combine cream cheese, confectioners' sugar, butter, and vanilla extract in a large bowl; beat with an electric mixer on low speed until smooth. Add milk, chocolate pudding mix, and butterscotch pudding mix; beat on low speed until blended. Fold in whipped topping using a rubber spatula.
Spread 1/3 of the cookie crumbs in the bottom of a 9x13-inch baking pan. Top with 1/2 of the pudding mixture. Repeat layers. Scatter remaining cookie crumbs on top. Chill until set, at least 3 hours.
Cut cake into 12 pieces; poke a gummy worm into each piece.