The wavelength at which a dune pattern emerges from a flat sand bed is controlled by the sediment transport saturation length, which is the length needed for the sand flux to adapt to a change in wind strength. The influence of the wind shear velocity on this saturation length and on the subsequent dune initial wavelength has remained controversial. In this paper, we present direct measurements of the saturation length performed in a wind tunnel experiment. Complementary to this, initial dune wavelengths are measured under different wind conditions --~in particular after storms. Using the linear stability analysis of dune formation, it is then possible to deduce the saturation length from field data. Both direct and indirect measurements agree that the saturation length is almost independent of the wind strength. This result supports the idea that grain inertia is the dominant dynamical mechanism limiting sediment transport saturation on dunes.