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, there are more publications; there's even a volume, but empirical ones are actually much scarcer on the link with religious aspects.)