After back-to-back wins for best picture at the Screen Actors Guild Awards and at Britain’s equivalent of the Oscars, the papal thriller “Conclave” has a chance at a hat-trick on Sunday at the Academy Awards.

The shot at cinematic glory comes at an awkward time: Cardinals and the faithful in Rome have been fervently praying in St. Peter’s Square each evening that life will not imitate art. Millions more are doing so around the world.

Pope Francis, 88, is more than two weeks into a stay at Policlinico Agostino Gemelli, a Rome hospital, for pneumonia in both lungs, along with other infections.

The Vatican said on Sunday morning — two days after a bronchial spasm that required him to undergo noninvasive mechanical ventilation — that the pope was resting after a peaceful night.

On Saturday, the Holy See had reported that Francis was stable and that he was alternating that mechanical ventilation with long periods of high-flow oxygen therapy.

But the crisis on Friday again renewed concerns about the prognosis for the leader of the Roman Catholic Church, and its own future.

The film “Conclave” has become a primer of sorts, giving millions a glimpse into the traditional — and secretive — rituals that regulate the election of a new pope.

The word conclave — from the Latin “with key” — refers to the isolation imposed on the cardinals who are locked inside the Sistine Chapel until a new pope is chosen, which is meant to keep the electoral process from dragging on. The cardinals take a vow of silence that can be broken only with papal permission (though leaks are plentiful).

The film stars Ralph Fiennes as Lawrence, the dean of the College of Cardinals, who in the movie is responsible for leading the papal election, and Stanley Tucci, John Lithgow, Lucian Msamati and Sergio Castellitto as papal contenders.

Their characters are not based on real people, but are instead amalgams of contrasting blocs, traditionalist and progressive, that loosely correlate with existing currents in the church.

Francis has tried to position the church to be more inclusive, giving rise to concern from critics who worry that he is sacrificing its traditions in the process.

He has allowed debates on previously taboo topics — like priestly celibacy, married priests and the extension of sacraments to the divorced. In doing so, he has set in motion subtle shifts toward liberalizing changes that have enraged conservatives as going too far and frustrated progressives as not going far enough.

A conclave will ultimately decide whether the church will pursue Francis’ vision or shift in another direction.

Alberto Melloni, a church historian who is writing a book about the history of conclaves, said that the film drew very precise fault lines but that in real life, cardinals would “not be so open about their antagonisms.”

“But to make a film,” he added, “you have to make explicit what in the real world is implicit.”

“Conclave” is hardly the first film to chronicle a papal election, but it may be the first where so much care has been taken to get the liturgical details right.

Robert Harris, who wrote the novel the film was based on, and the scriptwriter Peter Straughan were able to reflect the precise rules that Pope John Paul II established in 1996 for electing a pope.

Scenes show the Sistine Chapel being swept for electronic listening devices; the Latin oaths the cardinals swear before and during voting; and the tradition of threading the paper ballots after they have been counted so they can be preserved.

For his novel, Mr. Harris said in an interview last year, he turned to the former archbishop of Westminster, Cardinal Cormac Murphy-O’Connor, as a consultant.

When the book was published, Mr. Harris recalled, the cleric wrote him “a very nice letter saying it was just like a conclave and that the research was very good.” (The cardinal died in 2017.)

As for the plot and its surprise ending, the cardinal noted, “it was only a novel,” Mr. Harris said.

The film’s production team also took painstaking care when it came to recreating the Sistine Chapel and the more mundane Domus Sanctae Marthae, the Vatican guesthouse built to house cardinals during conclaves that is currently also Pope Francis’ home.

The production designer, Suzie Davies, who is up for an Oscar along with the set decorator, Cynthia Sleiter, said she had approached the film “more like a ’70s thriller than a religious film,” blending tradition, history and real-life mundanity like “cardinals on cellphones or vaping or going through security machines,” often set against the architecture of 20th-century Rome.

And the cast and crew had some insider training. The Rev. Elio Lops, the parish priest of San Vitale al Quirinale, one of Rome’s oldest churches, first gave the actors a grounding in Catholicism.

“They got a two-month crash course,” he said, including how to make the sign of the cross and to pronounce Latin prayers correctly.

Mr. Fiennes, who is up for best actor, was taught how a cardinal might behave in a particular situation. “It made them all very realistic,” said Father Lops, who also advised on Paolo Sorrentino’s television series “The Young Pope,” and on Fernando Meirelles’s 2019 film, “The Two Popes.”

The Vatican has made only passing mention of the film up to now. A short review in the Vatican newspaper L’Osservatore Romano gushed over Isabella Rossellini’s turn as Sister Agnes and acknowledged the “spectacular” nature of a conclave with its “rituals and myths.”

Cardinal Anders Arborelius, the bishop of Stockholm, said in an interview that he had watched “Conclave” on a plane to Singapore. “It was a bit exaggerated in some sense, but they were good actors,” he said. “It was interesting to see.”

Emma Bubola contributed reporting.