Preheat oven to 350 degrees F (175 degrees C). Grease a 9-inch square baking dish.
Mix oats, brown sugar, all-purpose flour, and whole wheat flour together in a bowl; cut in butter until mixture resembles coarse crumbs. Press half the oat mixture into the prepared baking dish.
Bake in the preheated oven until crust is set, about 12 minutes. Set aside to cool.
Beat cream cheese and white sugar together in a bowl using an electric mixer until smooth; add egg, lemon juice, lemon zest, and vanilla extract. Beat mixture until smooth; fold in blueberries. Pour filling onto the baked crust; top with remaining oat mixture.
Bake until bars are lightly browned, about 30 minutes. Cool in refrigerator, 1 to 2 hours.