Global High Temperature Spiral from 1880 to 2021, seems like after atomic weapons temps accelerated
First: Chernobyl wasn’t a nuclear explosion. It was a steam explosion that threw nuclear material around. Like a steam-based dirty bomb.
With that out of the way: technically, the two A-bombs dropped on Japan have had an effect that can be detected today, but only because we’re really, really good at detecting radiation. A nuclear weapon being detonated today would not be noticed atmospherically unless you were specifically looking for one (or in the vicinity of the blast, obviously).
The larger nuclear-related effect, though still requiring specialized instrumentation to pick up, comes from the continuous nuclear testing throughout the Cold War (~20-50 nuclear detonations per year ~1951-1990).
The difficult part of that is separating out the radiation from nuclear weapons from the radiation from coal plants (yeah, coal plants give off tons of radiation, surprisingly)