"Let the Ants Try" by Frederik Pohl [as by James MacCreigh]

 First appeared in Paul L. Payne's Planet Stories, reprinted by Pohl himself, Robert Silverberg, and Algis Budrys. Read online.

Summary:

After a nuclear war, the radiation creates ants with lungs. Two men, including Dr. Salva Gordy, see this as an opportunity to go back in time.

Discussion with Spoilers:

This seems to be Pohl's mash-up between H.G. Wells' The Time Machine and his "The Empire of the Ants." Without lungs, there is a limitation to the size an insect might grow, hence the spontaneous of lungs in ants--however improbable.  It may still be that mash-up, but apparently it began as several Midtown Manhattan discussions between himself and George R. Spoerer (not known to be a writer--the only info that I can seem to find is that he was 16 years Pohl's senior and had an apartment at one time in Brooklyn's Jackson Heights). Another source points out stories that precede this one as the inspiration, but Wells' supercedes them all and being the more famous and more reprinted, perhaps a more likely source. Wells, after all, is mentioned in the story.

The idea is to take these ants back in time and release them and occupy the human beings throughout history in wars with the ants, thereby preventing nuclear war. 

This seems a half-baked idea, especially how far back in time he goes. More likely, three scenarios would occur: 1) The ants eradicate the humans, 2) the ants are still battling humans, perhaps preventing educational/technological progress, 3) one species domesticates the other into companions or as a beast of burden, 4) the humans eradicate the ants.

Only in scenario 2 is it possible that the nuclear bombs are not created. Apparently, the ants can manipulate equipment and reverse engineer technology (although presumably using different appendages), so they seem just as likely to build a nuclear bomb at some point, making the whole project foolhardy. 

I had assumed, at first, that the clip below was the goal, thanks to the title. So I was surprised that the protagonist cared what the ants did to the humans since humans had destroyed the world (whatever the ants did would have to be better), but rereading I paid better attention to how he simply wanted his family back. Note the first "Salva" which in Spanish means he/she/it saves (although that could have referred to the ants as well.)

Therefore, apparently, the unnamed narrator titled this story--as well as having written it since who else was left to write it? But of course that means they went back to different timelines to witness events and assumed what other people were thinking.

Pohl himself was the first to reprint his  own story (Silverberg followed 20-odd years later). I'll let the cover of that anthology speak for what he thought about his own story. He said it was the first story of his he thought worth preserving

In the 1977 movie adaptation of Wells' "The Empire of the Ants," characters welcome their new ant taskmasters which The Simpsons famously allude to where they stirred up a number of memes, decades later: