Is there a "more formal" (or ideally, actual medical/research term?) for being "clucky" / "broody"?

I found the term reproductive ambition. You would need to delve into the literature to know how much/quickly it can change because of environmental factors.

Additional questions were asked about maternal personality (importance of having children, self-rated maternal/broodiness), reproductive ambition (ideal number of children, ideal own age at first child) and career orientation (importance of having career). Higher circulating testosterone levels were associated with lower scores on measures of maternal personality and reproductive ambition. D. K. Deadly et al.; "Maternal personality and reproductive ambition in women is associated with salivary testosterone levels", Biological Psychology (2006)

The current study shows that preferences for male faces with masculine shape cues, characteristics that are thought to signal men's underlying health, are positively correlated with partnered, but not unpartnered, women's reported reproductive ambition (i.e., their desire to become pregnant). C. D. Watkins; "Reproductive ambition predicts partnered, but not unpartnered, women's preferences for masculine men", British Journal of Psychology (2012)

Sometimes only the Abstract of a journal article is available for free; other times the entire Full Text is free. If a Full Text exists, there should be a link to it on the page with the Abstract. If there is a pay wall, there may be a rental option that is usually much cheaper than the purchase fee. You may be able to get free access at your work place or an institution, such as a local or medical library. Some sites like www.sciencedirect.com have free registration that gives you access to the Full Text of articles in participating journals.

IMO the best places for this type of research are PubMed and Google Scholar.