I think a big problem with comparing this is that monks and nuns have lives that are quite different from the Amish. Access to healthy food, modern medicine, and other non-luxury aspects of modern society are available to monks and nuns, whereas the Amish might choose to eat a diet with unhealthy contents without really questioning it due to their closed communities, and they might choose to avoid certain types of preventative medicine (including vaccines, which are generally taken by children before starting public school and prevent a lot of childhood diseases that are uncommon in the modern world).
It would be interesting to see what the differences are, although I imagine that they would largely show a health benefit to life in a religious community. Unhealthy aspects of secular life would be more difficult to maintain, and psychological or physical problems would probably be obvious in a tight-knit community.
The nun study, though, doesn’t really seem to answer the same questions that the OP was asking, since it only used nuns as a sample group for reasons unrelated to their lifestyles, and since it wasn’t testing correlation between lifestyle and disease. From the Wikipedia article, I understand that it was testing for traits that could predict alzheimer’s disease, and used nuns as a population group because they all had a similar environment and similar lives.