In Germany, the early predecessors of the green party were interest groups that addressed topics they felt were not being addressed by federal politics. The 1968 student protests and the subsequent subcultures had a major impact on this taking place in the 70s. The major clusters in the early green party were the anti-war movement, different ecologist movements, esoterics and in particular anti-nuclear groups.
One of the earliest public attention events in regards to anti-nuclear protests in then West-Germany, was a protest against a planned nuclear power plant in Wyhl, Baden-Württemberg, from 1974 onwards. There were widespread protests organized by alliances of local population, most prominently farmers, and what we would today call activits.
This period conicodes with some major chemical accidents, as a 1969 Endosulfan accident which killed large parts of the Rhine's marine life. The nuclear plant was also supposed to be built on the Rhine and the protest alliances were concerned about more of what the perceived as unforeseeable consequences of human hubris. Keep in mind that this was a big part of late 60s and 70s pop culture, with things like planet of the apes, spaceship earth, the club of rome etc.
The protest groups were successful in their endeavor to stop the construction of the power plant after tens of thousands of signatories signed plebiscitary lists and legal battles were won. This had a major publicity impact.
A similar grass roots protest to mention is the actions of local residents at Gorleben against a nuclear waste interim storage facility in their vicinity. This is an ongoing protest that has started in 1978, and has since that year seen heavy politicization with green-predecessor candidates winning on that platform, which gained national attention.
I do not have detailed knowledge about early green parties outside Germany, but would argue that the party has probably had at least some influence on its neighboring affiliates.
The obvious confirmation and fortification events of this ideologized and philosophical resent towards nuclear energy being the Three Mile Island incident, Chernobyl, and later Fukushima.
In short: party members were never particularly fond of anything nuclear (also because of not separating civilian and military use, while being also born from pacifism), having grown up in a period in which Hiroshima was widely viewed as attrocity, the nuclear annihilation threat of the cold war was permanently looming and mankind's impact on nature was noticable in your neighborhood. Then add Chernobyl and it's a very solid narrative.
A very interesting answer by /u/1upisthegreen1, let me add a little more information that's not specific to Germany. The modern environmental movement emerged in the late 60:s and early 70:s depending on the western country. To understand why the green movement was and is so critical of nuclear power, we have to understand the world view of the environmental movement during this time.
A good starting point for understanding the early modern environmental movement is Rachel Carson's book Silent Spring, that's published in 1963. Carson's book, which describes the dangers of chemicals in our environment and specifically the risks of DDT, becomes a smash hit and makes many people view nature and industrialization in a different light. Some people have argued that this book even should be seen as the starting point of modern environmentalism, allthough that can be disputed. Regardless, the book, even though it discusses the risks of chemicals in our environment, provides a very good starting point for understanding the ideas and values that leads environmentalists to reject nuclear power.
The most revolutionary thing that the book did was to introduce the idea of ecosystems as global to a large audience. The idea that we are living on a single planet and even pollution that takes place far away can have serious effects on humans. Well before the 60:s, there had been conservation movements in both the US and in Europe. These early environmentalists tried to protect specific areas from human exploitation. In other words, they saw environmental problems as a local problem. What Carson showed was that the problems of DDT were not local, but risked effecting even ecosystems far away from where the chemical had been sprayed, and did not just affect nature but also human health. This new global view of environmental issues, where we realized that the impact of industrialization was so great that we could effect not only local areas, but whole countries, and possibly the whole world with our activities, was a turning point for the environmental movement.
The second piece of the puzzle to understand the green skepticism towards nuclear power is also present in Silent Spring, and it's the backdrop of the cold war, and the fear of nuclear war. Throughout the 60s, fear of nuclear annihilation was very real among Americans. In Silent Spring, Carson keeps comparing the spread of DDT in our environment to the spread of plutonium over Japan that took place after the bombings of Hiroshima and Nagasaki. The fear of nuclear annihilation was another example of how people during this period are becoming aware of the incredibly destructive effects on the entire planet that industrialization can have, if not used correctly.
For the environmental movement on the 60s and 70s, their approach towards new modern, global problems is what united them. To us today, it might not be obvious what connects anti war-movements and environmentalism, but during that time they were very closely related. They worked against the cold war arms race and nuclear bombs. They worried about overpopulation and a lack of resources on the planet, and they viewed pollution as something that could have global consequences. Nuclear power was a new and relatively untested technology, that was linked to production of nuclear bombs. And a failure would have huge impact on humans and nature alike over large areas. Thus, it was no surprise that the environmental movement would reject the technology.
Even more could be said here about the green movements tendency to prefer local solutions to large scale ones, and it's skepticism towards technology in general, but I think that the interconnectedness of antiwar-movement and the nuclear movement via their global outlook on the world, as well as the fear of nuclear material, bolstered by the zeitgeist of the cold war, provides a pretty good explanation.
Main sources for this text is William Souder's "On a farther shore" a biography of Rachel Carson, as well as "Den gröna vändningen" (The green turn) by David Larsson Heidenblad.