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There is no boundary layer separation on a flat plate because there is not an adverse pressure gradient reversing the flow of motion.

A boundary layer is a thin region of a fluid, near a surface, that is flowing more slowly due to friction between it and the surface. Separation occurs when that thin layer is disturbed by a force acting in the opposite direction of flow. This force is called an adverse pressure gradient and is generally caused by the fluid moving more slowly downstream. On a flat plate, the fluid only slows down inside the boundary layer and is generally unaffected downstream.

If you were to tilt the plate up, the fluid would flow faster over the top of the plate and then slow down as it moved farther along. This would create that adverse pressure gradient and eventually cause boundary layer separation. The same thing happens to a plane when it stalls. If you tilt the wings too far upwards, you get boundary layer separation and the wing stops generating lift.