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.