I think a lot of this varies by diocese and what the bishop permits, and how far the priest will push. DW and I married 29 years ago this week. She’s a cradle Catholic, but wasn’t a practicing Catholic at the time we wed. I was baptized and raised Lutheran but was a pretty hardcore agnostic at the time we wed. DW didn’t want to be married in a Catholic church with a full Mass and all the traditional trappings, but she was still Catholic enough that it was important to her to have our marriage recognized by the Church. As it happened, she knew some pretty progressive priests in influential positions in the archdiocese. We met with the priest who made the recommendations to the archbishop on these matters. He was pleased to hear that I was baptized Christian, saying that wasn’t an absolute requirement, but it made things a lot easier. Even better, I could produce my baptismal certificate to prove it. His only real requirement was that I would not object if DW wanted to raise our children Catholic. I agreed to that, knowing it still meant something to her. And that was pretty much that—no further religious tests or qualifications, no mandatory instruction, nothing. We were married outdoors in a beautiful ceremony officiated by two dear friends of DW, one a Lutheran minister and the other a Southern Baptist minister (from a progressive wing that had broken with the Southern Baptist Convention after it was taken over by hard line conservatives). Our marriage was registered with the Archdiocese and has been considered valid by the Church ever since. DW has been considered a Catholic in good standing, and she even started attending Mass again once she found a progressive parish more in tune with her thinking.
Fast forward 27 years. Our eldest daughter, baptized and confirmed Catholic, wanted to marry a man who was raised Quaker, never baptized, not sure he considered himself Christian. They met with the parish priest who wasn’t concerned about the baptism thing or the prospective groom’s beliefs; the priest was more concerned with whether their intentions toward each other were honorable, and whether DD still considered herself Catholic, which she did. From there it was smooth sailing. I think they might have been required to attend three classes. They were married in our parish church, at the altar, by the parish priest, but with the same Lutheran minister who had married DW and me co-officiating. I think the Catholic priest was initially uncomfortable with allowing a Protestant minister— and a woman, no less—to preach. But the sophistication and depth of her theology, coupled with her her deft pastoral touch, blew the priest out of the water, and afterwards he couldn’t stop gushing about how honored he was to have shared the pulpit with her.
I’m sure many priests and many bishops are more rigid and old-school. But there are places in the Catholic church where this isn’t so difficult.