Again, take a look at the (linked) Effects of Nuclear Weapons, which discusses the estimated and observed results of nuclear weapons. (Although the last edition was published in 1977, we haven’t performed any above ground testing since well before then and thus, have no new pertinent observational data.) While this book was in publication it was widely available, and in fact, you could order it directly from the Government Printing Office or purchase it out of their stores. There is plenty of other research and information on nuclear weapon effects; check out the Federation of American Scientists archives, for instance. The one area of nuclear weapon effects on which there is still significant classified research are high altitude electromagnetic pulse (HEMP) effects, and the primary reason for that is that most of those assessments are based on models and tools that, while largely obsolete, are still themselves classified.
The details of specific weapon systems and especially protection/shielding systems are often classified, but the basic effects of the energy release are pretty much available to anyone with knowledge of heat transfer and atmospheric dynamics.
I haven’t performed a bunch of “back of the envelope first principles calculations” because the effect of a saturated atmosphere or precipitation will vary significantly with the size and altitude of the device, as well as the ground effects (in the case of a low altitude weapon). The distribution of fallout, for instance, will depend on the yield of the device, the altitude at which it is detonated, and the height of the tropopause at that point; if material is ejected into the stratosphere and be carried much further than if it is trapped in the troposphere. In the case of precipitation, it depends on the energetic yield and the absolute amount of water in the atmosphere. Obviously, the greater density of the atmosphere will reduce the acoustic shock, and the more water there is will absorb energy. How much attenuation this provides depends on a number of different parameters, so there is no “back of the envelope calculation” that gives some kind of overarching insight.
Comparing the effect of a blizzard in blocking constant sunlight (at a temperature of ~6,500 Kelvin at the upper atmosphere) to the effective temperature of a nuclear weapon (several tens of million Kelvin at the initial surface of the x-ray sphere for an instant) is like comparing apples to Eskimo; not only are they not the same thing, they don’t even have anything like comparable effects. The sun doesn’t create giant shockwaves or heat the lower atmosphere up to mass dissociation; a nuclear weapon doesn’t continue to radiate for an indefinite period. Sunlight typically doesn’t melt falling snow because the ambient conditions are too cold; the thermal pulse and blast wave would most certainly vaporize snow out to some distance which depends on yield.