It could be very, very bad indeed.
The majority of radioactive releases from Chernobyl did not come during the actual explosion. After the explosion, the graphite in the reactor core caught fire, creating a huge rising column of hot air which carried uraniumn and fission products out of the core and high up into the atmosphere.
This reactor fire burned for 10 days. Attempts were made to seed the radioactive clouds and trigger rainfall, to bring down as much particles as possible before the clouds passed over Kiev. Cesium was spread all over Europe. Eventually the fire was put out, ending the major source of contamination.
During this time, just a small percentage of its radioactive inventory was released. To prevent further leaks, a giant concrete sarcophagus was rapidly constructed over the disaster site. This cleanup operation took 600,000 people, and more or less bankrupted and ended the Soviet Union.
Now consider what would happen in the case that several (or 433) of these types of disasters occurred simultaneously. There simply isn't enough resources (people, machinery and economic) available to stop the disaster. So we could very roughly imagine each reactor burning off most of its inventory producing a disaster an order of magnitude worse than Chernobyl.
However, on the plus side, Chernobyl, being a rather old and crappy design, didn't have any major containment systems designed to prevent massive release. They simply never considered it at the time. But reactors constructed since have been severely regulated and safety systems improved.
Still not fail-proof, obviously, since Fukushima suffered simultaneous meltdowns. The amount of radiation released is highly debated, with estimates from no radiation related deaths, just a few cancers, to the entire north pacific being annihilated. What is known, is that enough radiation was released to merit widescale topsoil removal, and that the reactors are continuing to leak contaminants into the sea even today.
It's important to note that Fukushima is an on-going disaster, and should be treated as such. Six years after the initial events it is still very unclear what has happened, where the fuel is, what condition it is in, and how much of it has escaped. No efforts to actually fix the meltdowns has taken place yet. Japan, supposedly a world leader in robotics, have sent a handful of probes in there, most of which malfunctioned and/or returned untrustworthy data.
The Japanese government have done very well to recover from the associated tsunami, which killed 150,000, destroyed so many homes and infrastructure, and was obviously a much worse disaster. It is very interesting to note that the same government has not been able to clean up the Fukushima situation in any meaningful way. The site still leaks, they don't know what happened to the fuel, and they have taken no real plan to fix the cores. For reasons both technical and political, the cleanup is likely to drag out for decades, so the total radiation release may yet be counted higher than Chernobyl.
If they cannot make any progress on cleaning up one disaster site, despite 6 years and tens of thousands of workers, I think its reasonable to assume that there will be an even poorer response to a multiple-site disaster scenario. A modern malfunctioning reactor might not violently burn off all its radiation to the atmosphere, but it may just continue to slowly leak and pollute indefinitely, with even wealthy and high-tech governments like Japan being totally unable to stop it.
Should many reactors be damaged and pushed into disaster situations, I think the governments will be totally overwhelmed and unable to respond, and the reactors will simply burn and melt uncontrollably until they eventually burn themselves out, producing far worse radiological disasters than we've seen with Chernobyl and Fukushima.
The released radiation will contaminate crops, soil and water pretty much everywhere, entering the foodchain, and hanging around for tens of thousands of years. It's likely civilization will continue in some form, but I expect a large degree of mutations, non-viable births, massive increases in cancer and leukemia, with a worst-case scenario involving the total collapse of the food chain due to some critical component dying out, and triggering a mass extinction event.