Yes and no.
Sure, the '60s was when Lovecraft stories first started seeing screen adaptations ("The Colour Out of Space" became Die, Monster, Die, "The Case of Charles Dexter Ward" became Edgar Allen Poe's The Haunted Palace.) But these were presented more as supernatural events, rather than atomic-age horror. Sci-fi of the era didn't have an obvious Lovecraft influence, although some things have a Lovecraftian feel, like Fiend Without a Face. And, in fact, this illustrates why atomic-age horror and eldritch horror are distinct: there tends to be a link in eldritch horror between "weird stuff happening now" and "weird stuff that has happened for untold ages", the discovery of which threatens the sanity of the heroes. In atomic-age horror, it's only the villains who are toying with their own sanity, and that's because they've crossed a line to use Science in ways it was never meant to be used.
The Fungi do have a bug-eyed monster feel, though, ad "The Whisperer in Darkness" could be adapted to an atomic-age setting: switch the scholarly research into folklore and Forteana into some kind of geological survey in search of ores for new alloys or fuels, have the Fungi (or a Fungi-like stand-in) sabotage the survey in pretty much the same way.
The most atomic-age parts of the story:
- the way they replace humans; and,
- what they do with the humans they replace.