The Fukushima incident is a good model for what happens when a nuclear power station loses power. At Fukushima the reactors safely shut down following the Tohoku earthquake, but this only means that the primary uranium reactions stopped. The nuclear fuel continues to generate heat due byproducts undergoing radioactive decay. This decay heat is significant, so power must be maintained to coolant pumps even after a reactor shuts down. If that power is not maintained then the shut down reactors can/will still get hot enough to melt.
All existing commercial reactors need active cooling after being shut down. If electricity stopped functioning then the coolant pumps would stop working, as well as their backups (which are diesel generators and batteries) and the reactor would quickly get dangerously hot.
The loss of electricity does not necessarily imply that the radioactive decay would also stop, as radioactive decay is a function of the weak nuclear force, while electricity is a function of the electromagnetic force.
As the reactor gets hotter the coolant (typically water) would boil off as steam, increasing the pressure inside the reactor. This pressure would be relieved through emergency pressure release valves, but without active coolant flow the nuclear fuel would soon become uncovered and uncooled. High temperature reactions in the reactor (with the zircalloy fuel cladding) would release hydrogen gas, which would also build up inside the reactor and would be released through the pressure release valve. This hydrogen gas could explode, as it did at Fukushima.
Ultimately, one of three things will happen. Probably the best-case scenario is that the reactor will get so hot that it completely melts and falls into whatever happens to be under the reactor (hopefully a large concrete catchment basin designed to do exactly that). This nuclear slag will stay very hot and very radioactive for a long time, but otherwise it's not doing much. As long as you don't go spelunking inside former nuclear power plants you're going to be pretty safe.
The middle-case scenario is that the plant develops radioactive leaks. Radioactive particles will get into the groundwater and into the environment, but massive radioactive contamination is avoided. People who are unlucky enough to live downstream or downwind will have an elevated lifetime risk of cancer, but radiation-induced cancers have a latent period of about 30 years, and the actual incidence of cancer from long term low-level radiation exposure is generally very low. (About 40% of the healthy population in the world today will end up developing cancer, and about 20% will end up dying from their cancer. The natural prevalence of cancer is very high among the healthy population, so low-level radiation leaks will change that picture to affect more people and younger people, but the underlying reality won't actually change.)
The worst-case scenario is massive contamination such as what happened at Chernobyl. A steam explosion or similar mechanism rips the reactor apart and spreads radioactive fuel over the nearby countryside. However, your mental image of Chernobyl is probably a lot worse than it really was. Even without cleanup, the majority of the radioactive contamination would have been limited to the area immediately around the nuclear plant (less than a half a kilometer). People who venture up to the reactor complex itself could easily get lethal doses that would kill them in days or weeks, but if you never go up to the reactor site then you could just treat everyone else like they're exposed to low-level contamination like in the middle-case scenario.
Now, most western-style reactors are built inside a building called a containment building. This is designed to prevent any major radioactive release even in the event of a total core meltdown. As far as we know this mechanism will largely work, and even if containment is breached, it is likely that those breaches will be leaks instead of catastrophic damage and massive radioactive contamination. However, not all nuclear reactors (notably those built in Russia and the former Soviet satellite states by the USSR) have this containment. The Chernobyl reactor did not have containment- the reactor building was essentially a large metal shed you'd find housing any other piece of industrial equipment. When the Chernobyl reactor had a steam explosion it ripped straight through the building. What this means is that western-style reactors are probably immune to the worst-case scenario given above without additional plot intervention.
As far as how big those exclusion zones would be- Fukushima eventually had a 30km exclusion zone, but that is a very generous exclusion zone designed so that people would be exposed to absolutely zero additional radiation. Aside from a few hotspots outside the reactor complex there was no significant threat of acute radiation syndrome that would cause radiation poisoning or death (the dose rates mostly recorded were in the range of 10-50 uSv per hour- well below the threshold even for long term cancer risk).