So, we have a photon going c, as they are wont to do. The photon travels exactly 1 light year where it is reflected 180 degrees and travels the same light year back to the start (our detectors). Is any time lost from the moment of being reflected? Wouldn't the photon slow even just a little bit from the time interacted with the mirror? That tiny instance it interacted with the mirror, it was no longer traveling in a vacuum, it was traveling in the mirror's material. Or, does the interaction with the mirror not even count as a change in medium for the photon?
For a perfectly conductive mirror, there is no time delay, and the incident wave is reflected back instantaneously.
However, a for a real mirror composed of a material with finite conductivity and dielectric constant, things are a bit different. There is, in general a small time delay associated with the fact that the incoming wave penetrates a certain distance (the "skin depth" for a conductor) into the mirroring material. The penetration takes the form of an evanescent (decaying, usually exponentially) wave that decays inside the material. For an oblique reflection, this can give rise to the Goos-Hänchen effect, whereby the beam appears to reflect not off the surface of the mirror, but rather a plane further in.
The speed change you are asking about is essentially the temporal version of the Goos-Hänchen effect. An electromagnetic wave is delayed, because it is effectively reflected from not from the mirror surface but from a parallel plane deeper inside the material. However, questions of speed for photons can be a little tricky, since each photon has to represent a finite but extended wave train. When the photon is moving in vacuum, there is no dispersion, and it is possible to define a unique speed. In vacuum, the phase velocity, signal velocity, and group velocity are all the same. In contrast, for the evanescent wave decaying beneath the mirror's surface, it is not possible to define a meaningful, real-valued velocity. So to be quantitative about how the speed is affected by the reflection, you need to specify exactly what you mean by measuring the speed, and different choices can give slightly different answers.