The Unknown triggered a nuclear incident in 1986 in both worlds. This created the "God particle" (Cesium-137 is what the Stranger calls it). This in itself does not seem to cause the apocalypse, though like every event in the series, is a necessary component.
Straightforward, but possibly unsatisfying answer:
The trigger for the actual apocalypse in each case appears to be opening the yellow barrels in the power plant. It almost seems like just leaving the barrels open for awhile is what triggers it, but I think it's more complicated than that. The only thing I am sure of is that opening the barrels (once by Clausen in Jonas' world, the other time by Aleksander in Eve's world) is definitely a component to the apocalypse in each case.
My own speculation (ignore this if you don't want interpretations of the series):
I want to speculate more on why I think it occurs at those times, and it isn't just as simple as opening the barrels. I think it's somewhat important that it requires more than just opening the barrels, since we actually see Jonas ("the Stranger") open a barrel in season 1 episode 9 on the trucks, and it doesn't cause an apocalypse.
I think if we look at the bigger picture, the two important factors are location and time.
Because of the cyclical nature of everything in the two worlds, the timing of the event may be almost predetermined. If it does not happen at the specific time in each world, then some events would not happen the same, and thus events wouldn't have led to that situation at all.
I think this is even further supported when we think of the locations of the events. The barrels only seem to trigger the apocalypse when opened from the inside of the power plant, in the same room on each world it seems. After the apocalypse, a remnant of the portal remains, which then gets manipulated for time travel by various people who lead to the apocalypse in the first place. The apocalypse also seems to tie in with the "loopholes" which get exploited a few times to create diverging realities.
In full speculation on my part, due to the cycle, there might be something special about opening the barrels at those specific locations and times which triggers the apocalypse.
It's a bit of a chicken and egg scenario all around. Does the loop cause the loophole, or does the loophole cause the loops/apocalypse in the first place? I don't really know if there's a clear answer, it seems to be a fundamental part of the cyclical nature of time travel that the apocalypses must occur at the times and locations they occur.