It is my general understanding that unless you are severely deficient, Vitamin D supplementation generally takes weeks to bring levels up. It's unlikely that taking it for a few days is going to have any measurable impact on your recovery from illness unless you are severely deficient and/or taking MASSIVE doses, which may or may not be recommended depending on your prior levels and BMI.
See more here: https://www.ccjm.org/content/89/3/154
e: fixed broken URL
Uhm, how can you get to that conclusion? I mean: how can you compare the evolution of a cold with and without the vitamin surplus?
That said, do not underestimate the health benefits of the placebo effect. It can help a lot. Particularly with anything to do with stress.