There are roughly 400 nuclear power plants in the world today. If disrupted for several weeks, they would eventually lose cooling capabilities, as grid, backup fuel, and ultimately cooking water, were exhausted. You'd see a few hundred core melt events scattered over the globe, and moreover, no real capability to address those -- Chernobyl was encased in a large concrete sarcophagus (and is in the process of being re-entombed). Fukushima has still not been contained and requires ongoing interventions to slow, not halt, leaking of radioactive material from four cores.
S.G. Collins, the guy who put together the "No, we couldn't have faked the moon landings" video, has a very good one on nuclear power which addresses a number of the systemic and very long-range issues involved in nuclear risks:
http://fixyt.com/watch?v=KOBEloymLQA
The requirement for civilization to continue operating in a sane and predictable manner for a very, very, very long time is particularly concerning.
First of all, Chernobyl Exclusion zone is about 1000 mile^2. [3] Life is not doing just fine there. And the effect of Fukushima meltdown are not going to be known for the next few years, but there are indications that it may not be as harmless as we are led to believe.[4]
There are currently 62 commercially operating nuclear power plants with 100 nuclear reactors in 31 states in the United States. [1]
A recent study led by European researchers found Fukushima is not alone, as 22 other plants around the world may be similarly susceptible to destructive tsunami waves, with most of them in east and southeast regions of Asia. [2]
The World Nuclear Association estimates that 20 percent of nuclear reactors worldwide operate in areas vulnerable to earthquakes.[5]
[1] http://www.eia.gov/tools/faqs/faq.cfm?id=207&t=3 [2] http://news.discovery.com/earth/plants/nuclear-plants-at-tsu... [3] https://en.wikipedia.org/wiki/Chernobyl_Nuclear_Power_Plant_... [4]http://fukushima-diary.com/2014/07/nuclear-expert-fukushima-... [5]http://www.frontiergroup.org/blogs/blog/fg/how-many-us-nucle...
"Some claim that the populations of traditional Polesian animals (such as wolves, badger, wild boar, roe deer, white-tailed eagle, black stork, Western marsh harrier, short-eared owl, red deer, moose, great egret, whooper swan, least weasel, common kestrel and beaver) have multiplied enormously and begun expanding outside the zone. These claims, however, are not substantiated by any systematic census of any animal taxon."[41] https://en.wikipedia.org/wiki/Chernobyl_Nuclear_Power_Plant_... [41] Møller, A P; T A Mousseau; F de Lope; N Saino (2008). "Anecdotes and empirical research in Chernobyl". Biology Letters 4 (1): 65–66. doi:10.1098/rsbl.2007.0528. ISSN 1744-957X. PMC 2412943. Retrieved 2012-02-07.
" All the fuel in all the reactors and all the storage pools at this site (1760 tons of Uranium per slide #4) would be consumed in such a mega-explosion. In comparison, Fat Man and Little Boy weapons dropped on Hiroshima and Nagasaki contained less than a hundred pounds each of fissile material."[1]
"Accordingly, the worst case scenario at Fukushima will not make the northern hemisphere uninhabitable and the total doom/ death of life on earth predicted by some is impossible. The most likely outcome would be catastrophic contamination of northern Japan forcing total evacuation of the population. The main global consequence will be serious contamination of the Pacific which may catastrophically affect Pacific biota." [2]
If "catastrophically affecting Pacific biota" would not cause the whole ecosystem to collapse,and make life impossible, I'm not sure what would.
If few nuclear plant meltdowns occurred at the same time, and there would be a problem with the cleanup effort (i.e EMP in the States) we could expect spent fuel explosions to release massive amount of radioactive material into the atmosphere.
[1]http://www.dcbureau.org/20110314781/natural-resources-news-s...