Is Heisenberg's Uncertainty Principle applicable to light?

No, it is not at least in the usual form (3D momentum-position inequality), because differently from massive particles there is no position operator for photons as is known from the theory of representations of Poincare' group and imprimitivity theory. See for instance Barut-Racza's textbook on representation theory. Extending the formalism some interesting results exist like this one. I do not know if this proposed re-formulation of HUP corresponds to experimental facts.
I do not think that Fourier analysis is enough, "without introducing $\mathrm{\hslash <!--\; \hslash \; -->}$", i.e., without quantum phenomenology, to say that HUP takes place for classical systems like waves of sound.
Yes, one could argue that if the Fourier transform of a wavepacket of sound has extension $\mathrm{\Delta <!--\; \Delta \; -->}{k}_1\mathrm{\Delta <!--\; \Delta \; -->}{k}_2\mathrm{\Delta <!--\; \Delta \; -->}{k}_3$, then the spatial extension satisfies $\mathrm{\Delta <!--\; \Delta \; -->}{X}_i\ge <!--\; \ge \; -->\frac{1}{4\mathrm{\pi <!--\; \pi \; -->}\mathrm{\Delta <!--\; \Delta \; -->}{k}_i}$
The point is that this spatial extension has not the meaning of the statistical uncertainty of the position of a "particle of sound" because no localization phenomena exist and also because momentum is not related to wave number vector through quantum relation $P=\mathrm{\hslash <!--\; \hslash \; -->}k$
Instead, if we are dealing with a Schroedinger wave of a massive particle and we perform a sufficiently precise (3D) position measurement, the particle localizes inside the support of the packet. In other words, a relatively small spot of space (also very very small with respect to the extension of the wave) is abruptly and discontinuously chosen by the system.
However, repeating the position measurement for identical waves, the position where the particle localizes fluctuates with a dispersion satisfying $\mathrm{\Delta <!--\; \Delta \; -->}{X}_i\ge <!--\; \ge \; -->\frac{1}{4\mathrm{\pi <!--\; \pi \; -->}\mathrm{\Delta <!--\; \Delta \; -->}{k}_i}$, i.e., $\mathrm{\Delta <!--\; \Delta \; -->}{X}_i\mathrm{\Delta <!--\; \Delta \; -->}{P}_i\ge <!--\; \ge \; -->\frac{\mathrm{\hslash <!--\; \hslash \; -->}}{2}$
These phenomenological facts are completely absent for classical waves, because they do not describe quantum particles.
Regarding photons the situation is quite complicated. As far as I know they localize on screens their wavefunction encounters during its evolution. But this is not a three-dimensional localization and it is difficult to handle this phenomenon with the standard Fourier analysis machinery (though we know that the three components of the momentum are related with the wave vector components according to the standard quantum relation $P_i=\mathrm{\hslash <!--\; \hslash \; -->}{k}_i$). The mathematical counterpart of this fact is the absence of the standard position operators along the three spatial direction. So it is difficult to state a precise form of HUP for photons.