In short: no.
EDIT 2: And the reason for this is how you phrase the question: "should the nuclear power plants remain unattended for longer period of time, they will simply overheat and cause major damage to their surroundings".
That(!) — the boldfaced part — does not happen, even in the very unlikely case of meltdowns. Not even in the worst case scenario of Chernobyl did we see that. Chernobyl is currently an unintentional wildlife preserve. Nature and wildlife are doing just fine, save for one part, known as the "Red Forest" where the fallout was so heavy it actually killed the vegetation. That however is a shining exception to the rule. The rest of the area — and Fukushima even more so — remain undamaged.
Then we can start to ponder what "damaged" actually means. "Damaged" as in "destroyed", "disfigured", "dysfunctional" or "disturbed"... no, that does not happen. But "damaged" as in "uninhabitable" or "economically unusable", that is different matter since humans are quick to abandon such areas.
However even with that definition you are going to have much bigger problems elsewhere. Chemical plants, refineries, oil wells, waste facilities and waste dumps, and — which is particularly alarming — dams of all sorts.... hydro power dams, tailings dams, coal ash dams and water regulation dams in particular. And the breakdown of clean water and sewage facilities is going to take a much higher toll on us.
And this is still while assuming that catastrophic failures do happen at under-staffed/abandoned nuclear power plants... a scenario which I am about to explain why it is not very likely at all.
The long answer: A nuclear power-plant can be shut down in seconds. Literally so. The issue then is residual decay and the heat that creates. And here is where it gets a bit curious and most people misunderstand.
Used nuclear fuel is not in a steady state where it remains at the same, let us call it "danger level", all the time and then - after a set time - click, it is suddenly turned off and stops being dangerous. Anti-nuclear campaigners aside, this is not how it works. Spent nuclear fuel starts off at insanely dangerous when you have just closed the reactor, to "worrisome" within a week, "handle with care" within a year, to "let's wrap this up and put it away" within 50 years... and then the rest of the time is pretty much just us being unnecessarily paranoid.
Spent nuclear fuel is like the embers of a recently extinguished campfire. At the beginning everything is crazy hot. But the hotter something is, the faster it cools off. So in the beginning, the activity of the fuel is intense. But the isotopes that cause this intense heat decay the fastest and therefore disappear quickly. The more time passes, only the less and less intense isotopes remain.
A quote from Blade Runner captures the essence of this:
"The light that burns twice as bright burns half as long"
The critical time where you need to keep actively cooling the fuel elements — in order that the residual heat will not melt them — is about one week. After that you need to keep them soaked but they will not boil away the water.
Within a month you can open up the reactor, take out the fuel elements and put them in the storage pool. Natural circulation is more than enough to keep the fuel elements nice and snug and safe to be around, as long as you top up the pool. Not because the water keeps them from melting but because water is an excellent radiation shield.
So with your scenario of 2 months until the apocalypse and 10-20% of the population remaning, nuclear power plants will have plenty of time to safe their fuel. If the worse comes to happen and some plants are left without the resources to safe them this thoroughly, they only need to keep the pumps going for a week, then leave it filled with water. The reactor pressure vessel and the containment will handle the rest.
In the long run we are then left with sites of spent fuel elements in pools, dry storage on the surface or in reactor vessels. Does that present radiation hazards?
Not really no. Sure there may be some local contamination from damaged fuel elements, but unless someone deliberately goes in there and starts lifting elements out of the pools and try to break them, the fuel cladding, the pools, the reactor vessels and containment buildings will keep the nasties — I-131, Cs-137 and Sr-90 in particular — inside. That is after all why they are there.
Sure... the "Irradiated Wasteland" trope is very popular and an effective plot generator. But if you are going for "reality check" here, then it will not happen with your scenario. If you desperately want to use it, the downfall will have to be much faster.
And — again — then your problems will be much larger elsewhere.
EDIT: In a pre-Fukushima scenario, then the disaster scenario where nuclear plants blow up left and right might have been slightly credible. Post-Fukushima however it becomes outright nonsense. Not only has Fukushima set the baseline for what a nuclear plant must be able to handle — a sudden and catastrophic loss of both cooling and emergency cooling — but other measures to mitigate the damage done by a meltdown have also been put into effect. Two of them are especially noteworthy...
Hydrogen re-combiners. Noted already during the Three Mile Island accident, accumulation of hydrogen in the containment is a major issue. That may end up exploding, as it did(!) during both TMI and Fukushima. The solution to this is to install passive hydrogen re-combiners. These are catalyzers in the ceiling that makes hydrogen combine with oxygen — in a non-explosive manner — and become water again.
Release filters/scrubbers. The other issue identified by TMI was the need to be able to vent containments in a controlled manner. This is actually rather simple to achieve, using scrubber pools and stone filters, which can absorb up to 99.9% of the substances of most concern.
Some counties had already after TMI started employing these countermeasures. Here is an example from Sweden, the Barsebäck Nuclear Power Plant. The cylindrical structure to the left of the two reactor blocks is the release filter, known as FILTRA.
So — again — the issue will not be with nuclear power plants because they are, compared to the rest of our civilization, ridiculously well prepared for disaster compared to some other problem areas.
A final note: as someone both invested in the debate on nuclear power and nuclear technology in fiction, I personally feel that it is long past due that we got over tripe such as The China Syndrome, or the TV series "24"...
"Eeep! Terrorists stole the McGuffin that controls all our nuclear power-plants and caused all 104 of them to start melting down, and we can't stop it unless we get the thingie back! Oh noes!! It happened in a few sites, and everyone around them are now dead!!!"
...because this trope is as silly and unrealistic as to assume a light drizzle over New York causes the city to disappear under six feet of water within a few hours. Nuclear tech in fiction has been assigned magical properties of the blackest sort for the past 60 years, and it is time we got over it and stopped using nuclear as a lazy plot-generating device in apocalyptic/dystopian fiction.