Another paper that has some empirical results from Australia:
The results showed that believers and non-believers did not differ in the belief in conspiracy theories. Unpacking this further though, we did find that the extent to which religious worldviews were endorsed predicted belief in conspiracy theories. Among believers, the importance attached to their religious worldview was directly associated with higher belief in conspiracy theories and this link was partly mediated by higher anti-intellectualism. [...] We find that it is not the self-categorization as religious, but the extent to which religious worldviews are endorsed that could predict people's beliefs in conspiracy theories.
The paper also somewhat predictably found that (in line with much other research) trust in political institutions was inversely correlated with belief in conspiracy theory. Besides the dry results, it also has this discussion bit as to how religiosity (and even belief in ideologies like Marxism) might share mechanisms with conspiratorial thinking:
What then is the link between religiosity and the belief in conspiracy theories? It is
clear that research evidence is not straightforward. On the one hand, there is indirect
evidence that religiosity is positively correlated with the belief in conspiracy theories.
Broader beliefs that support conspiracy theories in general are assumed to underpin the
tendency to believe in conspiracy theories (van Prooijen & Douglas, 2018). Beliefs in God
that have been connected with perceiving the world as a place where everything is
planned and controlled (Laurin, Kay, & Moscovitch, 2008) might well serve this function.
Indeed, previous research has linked religiosity to higher conformity and security values
(Schwartz & Huismans, 1995), stronger conservatism and traditionalism (Norris &
Inglehart, 2004), and higher political conservatism (Esmer & Pettersson, 2007). As there is
no room for coincidence, random events are threatening and seen as a consequence of
secret actions performed by a group of malevolent people (Imhoff & Bruder, 2014), and so
religious individuals with high religious identification may be more inclined to endorse
conspiracy beliefs than non-religious individuals.
On the other hand, recent studies seem to suggest that both more conservative and
religious and less conservative and non-religious individuals may adhere to conspiracy
beliefs for different reasons (e.g., Farias, 2013). This notion represents the functional view
on conspiracy beliefs and the so-called belief replacement hypothesis, according to which
people are naturally predisposed to believe and that those who reject religion, intuitively
choose something else to replace it with (e.g., Newheiser, Farias, & Tausch, 2011). Based
on this view, atheists too, whether explicitly or implicitly, espouse various types of beliefs
that are meaningful, help them to explain the world and, ultimately, can play a
compensatory role in dealing with adverse circumstances. According to Newheiser et al.
(2011), Existentialism, New Atheism, Humanism, and Marxism are examples of such
beliefs systems endorsed by atheists, but also less clearly structured beliefs, like
conspiracy theories, can appeal to atheists.
(Actually on this kind of theorizing angle, there are more publications; there's even a volume, but empirical ones are actually much scarcer on the link with religious aspects.)
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 everyday life from the prescriptions of their religion etc.
I've not read the full paper in this case, but the above finding from Australia seems confirmed in another study:
Belief in conspiracies is not restricted to the fringe dwellers of society. International research suggests that such beliefs are quite common and that conspiracy theories may serve three basic psychological motives (i.e., epistemic, existential, and relational) for individuals. Yet, little is known about conspiracy theory awareness or belief in Australasia. We report the first large systematic investigation of system-justifying motives using two nationally representative samples of Australians (n = 1011) and New Zealanders (n = 754). Our findings show that almost all are aware of local and international conspiracies, the majority endorse one or more, and that all three psychological motives consistently relate to conspiracy belief, but not to awareness. In a series of hierarchical multiple regressions, we find that relational (i.e., increased anomie and disillusionment with the government) and existential motives (i.e., less trust in others and increased religiosity) are uniquely and relatively more important than epistemic needs (i.e., decreased analytic thinking) as predictors of increased local and international conspiracy belief. Findings are discussed in terms of the importance of understanding conspiracy theories as an ideological belief system that may function to serve underlying psychological motives.
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/)
There's also a more limited study from Italy, which only inquired about a specific (Covid-19) conspiracy theory in relation to religion:
Religion and conspiracy theories could show both similarities and dissimilarities. First, as alternative religiosity and conspiracy theories tend to spread knowledge stigmatized by the authorities, we expect that alternative religious beliefs are positively associated with conspiracy beliefs. Second, as religion and conspiracy theories explain events with the agency of invisible forces and detect patterns in nature, also conventional religious beliefs are supposed to be positively associated with conspiracy beliefs. Third, church attendance is hypothesized to discourage conspiracy beliefs, as exposure to religious authorities could deter the adhesion to unofficial narratives. By employing data coming from the Italian joint edition of the European Values Study–World Values Survey 2018, the paper tests the hypotheses by analysing the association between the multiple dimensions of individual religiosity and belief in a conspiracy theory on pharmaceutical companies. Results show that, after controlling for confidence in political and religious institutions and attitudes towards science, only alternative religious beliefs, here measured by belief in the reincarnation, are positively associated with belief in the big pharma conspiracy theory. Empirical evidence suggests taking caution when looking at similarities between conventional religiosity and conspiracy beliefs.
Given that the Catholic church has been pretty supportive of the official state messaging on Covid-19, including on vaccines, I'm not too surprised of this finding.