If you're curious as to some methods details in this Australian study, anti-intellectualism was measured on an ad-hoc scale mostly dealing with questions regarding trust in experts and the value of (higher) education. Belief in conspiracy theories was fairly standard in asking about whether the official explanation for things like 9/11, the death of Lady Diana or the assassination of JFK (interestingly perhaps, no mention of global warming, even though that's sometimes centrally featured in contemporary US research on conspiracy theories.) The intensity of religious belief was measured with questions like how often the subject departed in every dayeveryday life from the prescriptions of their religion etc.
(This This one is probably more interesting/convincing as it used a nationally representative sample, not often done in psychology research. It also used a fairly "thick" index of 8 "international" conspiracy theories, ranging from 9/11, aliens, jfk, chemtrails, vaccines, to global warming, asked as separate question... plus seven "local" ones, like in the case of Australia, about the Port Arthur shooting, the death of PM Harold Holt etc. There's a preprint of this paper freely available at https://osf.io/4pnxy/)