Variation in cosmic ray flux causes changes in the formation of clouds in the atmosphere, by affecting the formation of droplets by charged nuclei, similar to the process of cloud seeding. The seeds are formed when high energy cosmic ray particles collide with other atmospheric particles producing a cosmic ray shower.
There are two main products of cosmic ray showers illustrated in the figure: short term products such as electromagnetic radiation, and the various products of collisions (called spallation) that persist, can form nuclei for droplet condensation, and can be detected later to provide useful information on the intensity of past cosmic ray showers.
The process is probably not the same as the rapid formation of clouds above and around atomic blasts, such as in the clip below. My guess is these clouds are formed by the blast wave. Still, it seems possible that a demonstration of high energy particles affecting cloud formation in the atmosphere should be possible.
Particles crashing into nitrogen and oxygen in the atmosphere create isotopes. Most collisions are with stable forms of the two most common gases, Nitrogen 14N7, and Oxygen 16O8. The most useful products are the unstable isotopes Carbon 14C6 and Beryllium 10B4, due to their half lives. 14C with a half life of 5730 years is useful over many millennia, but not so accurate at the long term. 10Be with a half life of 1.36 million years is useful for longer, but not as accurate in the short term.
Typically materials are aged by comparing the ration of the unstable isotope to the stable one. Over time, the unstable isotope is a lesser fraction of the total. However, there are factors that can cause errors. In the case of 12Carbon, because fossil fuels are low in 14C, present day deposition of 14C is anomalously low, so this needs to be taken into account when dating material less than a few hundred years old. Also, due to the difference in atomic weight, plants preferentially absorb the lighter stable 12C form, thus changing the isotopic ratios.
However, isotopic rations are an incredibly useful tool in studying the past because of the capacity to precisely quantify isotopic ratios, and helps to provide greater confidence in the causal role of cosmic ray flux in past climate changes. While the sediments where isotopes are recovered are still subject to confounding factors, my impression is that biological indicators such as tree ring width and density have even more uncertainties associated with them.