What if in an alternate universe after the Chernobyl incident, the world governments decide to shut down their nuclear power plants and if so how would that affect society?
Chernobyl was an RBMK type reactor. The RBMK is pretty unique in terms of reactor designs. Its advantages are that it is large with high power output, and it can be refueled while operating. Its disadvantages are positive coefficient of reactivity (explained below) and its large size allows different neturon fluxes to persist in different parts of the reactor.
Coefficient of reactivity is probably the most important concept to understand with a nuclear reactor. The Russian RBMK is one of the only designs with a positive coefficient. All the Western pressurized water and boiling water reactors have negative coefficients. The coefficient of reactivity is the ratio of change in reactivity to change in temperature. If the coefficient is negative, then as the reactor heats up, fissions happen more slowly. A negative coefficient reactor that is undergoing a high power casualty will shut itself down before it does too much damage. With a positive coefficient, a high power casualty will create a runaway condition where higher power generates more heat which increases power. You can see how this lead to massive explosion and containment breach. For American reactors, which are all of the pressurized or boiling water types, a Chernobyl type accident is physically impossible.
I tell you all this to inform you that the powers that be in US Nuclear power (the Nuclear Regulatory Committee in particular) will understand this. There is no way that the Chernobyl accident could happen in the US. Thus, it is very unlikely that Western countries, operating much safer reactors would shut down due to an accident in the poorly designed and poorly operated Soviet plant at Chernobyl.
Now Fukushima was a different story...
One of the things I think it would take for this to be realistic would be that in this alternate universe, the Chernobyl event was far more devastating in terms of fallout.
The second thing would be that the Chernobyl reactor was well maintained and of a solid design, but they couldn't figure out why it happened. I think these reasons would be the catalyst for the anti-nuclear camp to completely dominate. "It just killed 2 million people and we still don't even know why, 15 years later!"
The obvious result would be that different sources of power would have to replace the gap left by the disappearance of nuclear. I'm not sure how much this would affect society, since nuclear technology isn't exactly in everything like the world of say, Fallout, for instance.
Shutting down some power plant overnight is going to pose a tremendous challenge on the energy management system of any country doing it.
Let's assume the country has no short term alternatives: it will be more or less the same scenario of the oil crisis in the '70ies.
Germany exactly did that after Fukushima. They shut down all their nuclear power plant and completely withdrawed from the nuclear industry, even Siemens which was a big player completely withdrew. They massively switched to coal based electricity. Germany has a lot of coal so it wasn't that bad from an economic standpoint. Modern coal factories don't produce too much smog so environmentaly it was that bad from the smog perspective. CO2 however is another story...