All the above answers aren't incorrect, but seem to ignore some pertinent facts:
The explosion didn't "cause" Superman to "absorb the life of living things"; the ionizing radiation (you know, the deadly kind that causes cellular mutations and damage by breaking the bonds between cells) overwhelmed him, and did some damage which brought him back to earth. Needing fuel to heal himself, but with the sun (the source of the non-ionizing radiation which feeds his powers) blocked, he ... somehow managed to "tap" into the raw solar energy stored in the flowers which their chlorophyll would use to convert the sunlight into bio-energy, effectively "Recharging" his own cells to a limited degree.
This was the recently released, Post-Crisis, purely solar powered, "not unbeatable" Jon Byrne version of Superman. For YEARS it had been establish that a full on, ground 0 nuclear strike would not harm Superman in any way....
from the 40s....
to the 50s....
into the 60s.....
All the way into the 80s, multiple megaton nuclear blast were something the guy could film casually, shrug off... or at the very least, walk away from intact and recover from in a short time.
Barring any very slim possibilities----- the presence of Kryptonite, him being affected by magic, his being severely weakened do to over exertion, or the radiation coming from some unique space element, for example---- most regular ionizing radiation, up to Gamma wave level, just doesn't slow down Superman for long. Now, part of the logic behind this was Superman's incredible molecular density; the radiation simply couldn't affect his molecular bonds because they were just too tightly compacted, on top of the regenerative powers the sun gave him.
Then came the Post-Crisis reboot, and all of Superman's powers came from the sun.
Consequently, his "indestructibility" wasn't just from his body and enhanced by yellow sun radiation any more, but totally supplied by it; his body was no longer composed of super dense tissue, but produced a "bio-electric aura" as a natural consequence of solar absorption which essentially acted like a "personal protective force field" which Superman could control nominally; this is why it was so common to see his skin-tight suit taking little damage, but rips and holes appearing often in his cape post-1986. Being so much weaker, it seemed logical, as well as useful, that Superman could take some damage from cell destroying radiation now, even if he recovered from it quickly.
But the only specific radiation which was supposed to totally negate his powers, and thus that field, was Kryptonite radiation...
However, this new version of Superman was also consequently weaker physically; he was no longer able to move planets with ease, and in fact struggled holding up continents. Byrne did this to make the hero "more human, flawed, and subject to injury", so that the stakes of his adventures would be higher, more "realistic" and engaging to the audience. This reboot happened in 1986, though; that's the same year that TDKR came out, so by this time, this "new" Superman's limits hadn't been explicitly tested yet. Miller---an avid Batman Super-fan and vocal dissident to Superman---- basically took advantage of this in his graphic novel, and used it as an opportunity to weaken Superman in-story enough so that Batman could make him look like a fool.
So on a meta level, the reason one dirty nuke explosion could affect this Superman so much was because the writer's hadn't yet established that he was immune (or at least highly resistant) to such harm. This writer in particular then used that loophole/ wiggle room to put Superman on "a more even footing" with a character he favored for a final showdown. That's also the reason Superman gained the ability to "absorb solar energy" from plants... a power he point blank did not have in either Post-Crisis, or Pre-Crisis canon; Miller could not outright kill the icon for his plot, so he settled for humiliating him in story, to some degree, and gave a heretofore unforeseen ability as the reason he survived.
- Superman , in the novel, never once thought to go to the Fortress to recharge, or just to fly up into space beyond the clouds and take in some sunlight to get his powers back to normal. The reason why such a common-sense approach wasn't taken was never addressed in the comic.
Now, to be fair, the Fortress didn't really exist at this time (it wouldn't be built in Byrne's stories til a year and a half later) ... but neither did the "plant absorption" ability! So there could have been a way around that to get Superman back to full power. The book never showed how "weakened" he was compared to his normal levels, and he was strong enough to fly (at super speeds, no less!) under his own power, take electrocution, keep fighting when kryptonite gas was shot at his face while dueling with Batman, and lift/toss a tank with ease.... but flying a few miles up into the upper atmosphere to absorb sunlight for the hour or so that he could hold his breath and get somewhere back to his normal levels seemed to be beyond him.
Again...this was likely done more for plot relevant reasons; in story, Batman himself said he'd never be able to take on Clark at his full strength. So in order to keep the stakes high in their fight, Clark still had to be recovering. The extent of the radiation poisoning had to be shown to be catastrophic in order for him to be a treat to Bruce, but not something this older Batman couldn't "reasonably" overcome. So him being "Shriveled up" is more a visual exhibition of how much damage he took. And because the skies were blotted out by the nuclear winter [which again isn't factual, as one 15-kiloton bomb would detonated in the upper atmosphere would not cause enough soot and ash to arise from the land to blot out the sun for So many days!]he was not able to get the direct sunlight he needed at ground level to fully recharge, and we have no idea how many fields, mountains, or oceans of plankton [some of the richest solar feeding plantlife] he would have had to absorb in order to get to even half strength.