For a short while; it’s proposed that it would have cooled our climate off slightly from suspended particulate matter scattering insolation. An interesting fact though, radioactive dust deposited fairly uniformly around the world, so we can still see this layer in ice sheets which helps narrow the window for dating “older ice”.
The internal degrees of freedom of a solid allow to convert incoming em-radiation to lower frequencies (read: light to heat). If that happens in the higher atmosphere, the now lower frequency radiation will not penetrate the atmosphere and thus less sunlight will reach the ground. Thus, solid particles (ash, dust) in the upper atmosphere are a coolant and big explosions tend to throw a lot of those into the atmosphere. Thats the idea behind the so-called nuclear winter after a potential nuclear war. However, the actual nuclear tests did not throw that much matter into the air compared to large volcanic eruptions. Thats why the nuclear tests are barely visible in the global temperature curve, while big volcanic eruptions are really seen as a dip in the temperature for a few years. Anyways, solids are always heavier than air, and thus fall back to the ground over a few years. Thus those impacts are short-lived and do not contribute to long-term changes.
What impact the nuclear test did have though was to throw a lot of radio-active isotopes localized into the air. We can track those and measure their concentration in different locations over the years to learn how these isotopes get distributed over the earth by winds. This helped us a lot to understand global wind currents and is vital in order to build realistic climate simulations. Especially C-14 contains a lot of information, since the concentration is determined by the interplay of three factors: natural C-14 generation from interations of C-12 isotopes with cosmic rays (which leads to a balance between regeneration and decay and thus a nearly constant natural concentration), an enhanced C-14 generation through the nuclear test (which then decays again over time) and the burning of fossile carbon (in which all C-14 is decayed, therefore not present). When we manage to disentagle those influences, we can learn a lot about the spread of pollution, the human influence on C02 concentrations and where all the CO2 that we produce ends up (plants, ocean, atmosphere). Hopefully, that knowledge will in the future have some influence on the climate because it might motivate us to act accordingly.