Because of the density difference between the cold air and the warm air, you will have a pressure difference as well, so you will have a “kink” (refraction) in the isobars at the frontal surface.

The isobars will be “refracted” towards the low pressure.

This occurs in all the layers affected by the front, so it will veer all the way from the surface to the top of the front.

The “catch” is in the first words of the question. “At the approach of a warm front ……”.

If there was a wind before the “approach” of the front, then the surface wind would have backed in relation to the free stream flow.

As the warm front approached, because of the explanation above, both the surface wind and the upper wind would have veered.

The angular difference between the surface wind and the free stream flow will remain.