Fluoride-related conspiracy theories have been around since the Forties, promoted by anti-communism groups like the John Birch Society. How did this conspiracy theory so completely influence health food culture that it is now nearly impossible to find fluoridated toothpaste at a health food store?

It sounds like you're mainly wondering how efforts against fluoridation got associated with both far-left-wing and far-right-wing at the same time. Both have been associated with it from the beginning so there wasn't some "transition" as you imply; it's simply that conspiratorial views can be reached by multiple avenues. The 1960 study Classification and Appraisal of Objections to Fluoridation identified 255 different objections although the claims can be boiled down to roughly

Fluoridation is not medically effective.
Fluoridation is not cost-effective.
Systems that provide fluoridation can fail.
Fluoridation is a health hazard associated with (pick a disease).
Fluoridation is socialized medicine / communism.
Fluoridation is a violation of individual liberty.
Fluoridation is part of a general conspiracy by either government to pacify the population, or corporations to get rid of chemical waste.

You'll notice that the alternative-medicine types might tend to 1 and 4, while opponents of big government might attach instead to 5, 6, and 7. Same goal, different motivations.

For an instance of the alternative medicine approach, consider Fred J. Hart, with a passel of "alternative" medical electrical devices in the 1950s but had run-ins with the government. Blood samples would be sent in with diagnosis sent by mail, but the FDA found his "medical" equipment was essentially equivalent to a doorbell and put an injunction for him to stop his services. He refused and was cited for criminal contempt in 1962. As part of his fight against the FDA he founded the National Health Federation, and part of the group's platform included anti-vaccine (including polio, and polio vaccine pushback was not significant at the time!) and anti-flouridation.

Other groups in a similar vein are the "National Nutritional Foods Association" (1936) and "Earth Island Institute" (1982).

Conspiracy theories are hardly just the purview of the right-wing; for example, one of the ones trotted out with fluoridation has to do with the Manhattan Project. The Strong Memorial Hospital in Rochester did do some (horrifying) experiments with plutonium in relation to human subjects. (Quoting the Atomic Heritage Foundation: "Between 1946-1947, physicians injected six patients with uranium with the research goal of discerning the minimum dose that would produce detectable kidney damage".) Once uranium hexafluoride was known to be a substance in use there were some fluoride experiments added to the docket, but note that there is a major difference between fluoride and the uranium version, which has one atom of uranium put with six atoms of fluorine (hence the "hexa"). As you can imagine, folks like the John Birch society were more focused on the "socialized medicine" part of the conspiracy and (being generally supportive of more bombs) less the derivative-of-the-atom-bomb angle. (Presidential candidate Goldwater had affiliations with John Birch, and while it might not be fair to say he "wanted nuclear war", he was hardly against it; I've written about this more here.)

So, there hasn't really been a shift in position, even through more modern times. The 1991 study Scientific Knowledge in Controversy: The Social Dynamics of the Fluoridation Debate -- using interviews with people debating fluoridation in Australia -- identified this issue as one with particularly monolithic blocks where it is very hard to change minds; interviewees claimed the evidence was completely on their side, and there was very little criticism of "inaccuracies, exaggerations, and simplifications".

This is still a current issue, especially regarding places other than the US, where the number where 50%+ of the population has fluoridated water is very small: Australia, Brunei, Chile, Guyana, Ireland, Singapore, Canada, Malaysia, Hong Kong, and New Zealand. The only places with 100% are Hong Kong and Singapore. Ireland is the only country in Europe that mandates fluoridation.

...

Lehr, J. H., Freeze, R. A. (2009). The Fluoride Wars: How a Modest Public Health Measure Became America's Longest-Running Political Melodrama. Wiley.

Martin, B., Groth, E. (1991). Scientific Knowledge in Controversy: The Social Dynamics of the Fluoridation Debate. United States: State University of New York Press.