If the popular-cultural phenomenon of Zombies came to prominence in the 60s, might they have origins in the Cold War?

Goes back quite a bit further, actually. Stories of resurrected dead go back centuries, but the current popularity, and especially its association with voodoo, comes from the 1920s and 30s with the first zombie fiction in the pulps and the first zombie movies, especially White Zombie (1932). To quote from Jeff Shanks in his introduction to Zombies from the Pulps:

The public's current fascination with zombies is generally seen as the peak of a cultural trend that began with George Romero's Night of the Living Dead (1968), continued through the 1970s and early 1980s with several sequel films and various knock-offs, faded somewhat in the 1990s, only to reemerge stronger than ever after the turn of the millennium. But there was an earlier "Zombie boom: that predates the current fad, predates Romero's groundbreaking fim, and even pre-dates the undead-filled EC horror comics of the 1950s. Zombies first entered the American public consciousness in the late 1920s and early 1930s, and for a little over a decade they were a frequent element in horror fiction both on the silver screen, on the stage, and in print, before fading into relative cultural obscurity until Romero reimagined the trope over two decades later.

We won't dwell on the '20s/30s zombie craze, since you're more interested in the '60s, where the key work is George Romero's Night of the Living Dead (1968). Romero's most direct influence was Richard Matheson's novella I Am Legend (1954), which introduced the idea that being "undead" was not a supernatural condition resulting from a sorcerer's spell or even the "living dead" zombie rendered a will-less servant by some potion, but the victim of a virus - one that caused a hunger for human blood (Matheson) or flesh (Romero). It was a turn from fantasy to sci-fi, and combined with the epidemic imagery, was startlingly effective.

I'm not aware of any direct influence that Cold War preparing-for-atom-bomb mentality might have had directly on the zombie film. But you can easily argue that the Cold War - the looming imagery of an apocalypse - was in part responsible for the rise of several such dystopian films in the 1960s, including The Omega Man (1971, which was actually adopted from a Mattheson novella, and is probably closer to what you're thinking of) and Planet of the Apes (1968), and the turn of the zombie film from purely supernatural horror to a scientific or pseudo-scientific model certainly fit in with the presiding atmosphere.

Outside of such direct influence, zombies have also found some ground for Marxist readings. If you crack open the Encyclopedia of the Zombie, there's an entire entry for Class by June Michele Pulliam:

Because the single most defining characteristic of the zombie is its lack of free will, it is a fertile metaphor for class, particularly the plight of the working classes and their lack of control over the means of production relative to the members of the middle and upper classes. Thus, class struggle is an underlying theme in most zombie fiction.

Pulliam isn't describing Soviet-style communism, but she does trace the mindless zombie concept back to its metaphorical roots in African slavery in Haiti, and you can definitely see the parallels between a slave rebellion and the siege-like mentality of Romero's outnumbered survivors, which gives for a nice Marxist reading - though I'm not aware of Romero ever intending that explicitly.