Here, from the little I have glimpsed, is my understanding of the connection between the contents of the two books.

The first, more famous book of course ends when the earth has been left with no humans on it, apart from the dead; and life forms are pretty much limited to the deeps of the ocean, humans having thoroughly laid waste to the land masses and the water within these. I recall a ship carrying the last survivors off the planet.

So, the second book, since it takes place on earth, cannot follow the first. Rather, the events of the book precede by some time the catastrophes that end the earlier book.

"Saint Leibowitz and the Wild Horse Woman" does give a lot of attention to power struggles within the Church, with characters intriguing to become the next Pope or to be the power behind the next Pope, and a lot of pages and chapters go into those developments.

However, what the second book also has is a character, not in the title, who embodies adolescent rebellion in a grown man's body; the Church, and the monastic life, are things for him to rebel against, and his rebellion is acted out over the course of the book. That means it gets raunchy in places. One character carried over from the earlier book is the wandering Jew. And the rebel I have just described, for all his conflicts and disruptive behavior, finds that he cannot cease to be a spiritual being even if he can't stand the Church: the wandering Jew, of all characters, becomes something of a role model. And that's about all I can relate, although it occurs to me now, that the story is post-atomic in that there are people with congenital defects assumed to be the result of radiation exposure.