For all practical purposes it's really completely unrelated to the speed of light. Nothing would change in the argument if light travelled instantaneously. Sure, the numbers would change by O(1e-6s), but I'll admit that I wouldn't be able to tell the difference when watching an aircraft.
I'd meant to edit my post to note that for the purposes of this phenomenon, light speed is instantaneous.
Though in the more general case, the phenomenon would apply to any case in which two signals or channels travel at different rates or speeds. Light and sound are the examples most familiar to us, though other alternatives exist.
Neutrinos can tell us what is occurring at the core of the Sun with an ~8 minute delay whilst the propagation of EMR effects from the Sun's core is thought to take 10,000 to 170,000 years, as these travel through repeated collisions, absorption, and re-emission.
Diffusion processes such as smell or other chemical materials both travel more slowly than either light or sound, and at different rates for different compounds --- heavier compounds diffuse more rapidly than lighter ones. This is incorporated into the chemical signalling processes evolved by insects such as ants, in which some compounds are heavy and complex (usually for food or other valuable resources), others are light and fast (danger or alert signals). Again, for a moving or propagating phenomenon, these will move at different rates.
For more complex phenomena, you might note that there are early / rapidly-moving indicia and those which move more slowly. Again, understanding the difference between these, the rates at which they travel, and their association and interactions with the processes originating and surrounding them will assist in drawing an accurate inference of the root phenomenon.
All sensation is mediated, not direct, and that mediation has a direct effect upon sensation.
