How to describe a person who does not want children?

Solution 1:

The common term these days is child-free:

child-free

ADJECTIVE

1. Not having any children, especially by choice.
   ‘a child-free woman who likes to travel’
   ‘my husband and I are child-free’

For example...

Ms. Handler’s and Mr. Dyer’s desire to be childless — or child-free, as some prefer — syncs with nationwide shifts over the last several decades...

Your sentence, then, would be something like:

When a unique post-apocalyptic baby is born, a retired, child-free headhunter becomes its guardian...

Solution 2:

"Without a maternal drive" might also work, though it is not a single word.

Maternal [muh-tur-nl] /adjective

1. of, pertaining to, having the qualities of, or befitting a mother: maternal instincts.

Source: Dictionary.com

Consider "drive", rather than "instinct", if you are concerned with the argument against an assumed innateness of maternal feelings. Or perhaps consider the term "nonmaternal", though I am not sure it is a proper word.

Solution 3:

There is a code phrase for this, popularized in the last century but still very recognizable today: career woman. From Oxford Dictionaries:

career

...
1.4 [as modifier] (of a woman) interested in pursuing a profession rather than devoting all her time to childcare and housekeeping.

Nowadays we don't think of a career and motherhood as mutually exclusive, but the phrase was coined to make exactly this distinction. Some examples of use:

[M]y wife had an aunt whose daughter was a career woman; she did not want children, and had never had a child . . . . —Fr. Anthony Zimmerman, "Interview: Lloyd J. Duplantis, Ph.D", The Linacre Quarterly, 1999

“She was a career woman. No time for children.” —Vivienne Wallington, The Last Time I Saw Venice, 2011

“Prospective employers can't legally ask if you want children in the near future, but you know for damn sure they'd like to,” she said. “So I say it up front. I'm a career woman with no time for children.” —Cathy Kelly, It Started With Paris, 2015

And a discussion of the anachronism of the phrase, which nonetheless acknowledges its underlying meaning:

The problem with the term “career woman” is that it's anachronistic; it's from a generation ago when a woman who worked was an outlier, a rebel, a feminist.
It's really not relevant to today . . . .
Yet even the Boston University Department of Economics couldn't resist the title “Are Career Women Good for Marriage?” for a 2008 report. The authors explained that by “career women,” they meant any woman, married or single, who works. Which only begs the question: Why are there no “career men”? And while there are certainly women who are child-free by choice (and sometimes that choice is made in order to have more career freedom), most of the women I know want children. —Melanie Notkin, Otherhood: Modern Women Finding A New Kind of Happiness, 2014

I would recommend using this phrase with caution, as it does not generally reflect today's reality and carries more than a whiff of sexism. However, if it works for your audience and story-line you could potentially take advantage of this relic of the battle of the sexes, with something like

When a unique post-apocalyptic baby is born, a former career woman becomes its guardian...

I would substitute former for retired here, just to avoid confusion about her age (unless she is actually in her seventies).

(Of course this phrase will work best if her back-story includes some kind of career that took up much of her time; if she was more a lay-about who happened to have a child phobia, you probably should look for something else.)

Solution 4:

childfree

The usage of the term "childfree" to describe people who choose not to have children was coined in the English language late in the 20th century. https://en.m.wikipedia.org/wiki/Voluntary_childlessness

It's not my favorite, because it can be confused for "child free," which could ironically make it almost its own opposite(child free and trying to change that).

My next line of thought is to go the opposite way and use a prefix to invert "someone who wants babies." "Broody" means that, but unfortunately it has other meanings and this one is lesser-known. Abroody, unbroody, disbroody, or antibroody?

broody adjective (WANTING CHILDREN) informal If someone, especially a woman, is broody, she feels as if she would like to have a baby: Much to her surprise, Ruth started feeling broody in her late twenties. https://dictionary.cambridge.org/dictionary/english/broody