NEW YORK — John Oliver will have the crab cakes – but that’s all, thanks.

In a private dining room at HBO’s plush headquarters overlooking Bryant Park in midtown Manhattan, a waitress in a crisp black uniform asks the 37-year-old comedian if he’d like anything else – perhaps some soup or salad?

He declines with a polite insistence that suggests he’s not quite used to all the luxurious trappings. Then, as soon as the waitress leaves the room, he launches into a riff about possibly sinister deeds going on behind the scenes at HBO.

“I don’t know what happens here,” he says, “This is, like, unsettlingly nice. I wouldn’t be surprised if it turned out to be a front for a meth operation. There are these beautifully padded walls and you think, ‘Oh, these seem soundproofed.’ But that’s never good when something seems unexpectedly soundproofed.”

Oliver should probably get used to his cushy new home. After more than seven years as a correspondent on “The Daily Show,” the Cambridge graduate is stepping into the spotlight with his own topical humor show, “Last Week Tonight,” which premiered last Sunday on the premium cable network. In a sign of his ascension to the top of the comedy ranks, Oliver now commands a prime piece of real estate on HBO’s marquee night.

Less than a year ago, Oliver was eagerly downplaying expectations as he prepared to step in for “The Daily Show” host Jon Stewart, who was about to take an extended hiatus to direct his first feature film. At the time, Oliver’s stated goal was simply: “Don’t have this building on fire when (Stewart) returns.” But during Stewart’s 12-week absence, the native of Birmingham, England, proved to be more than just a diligent housesitter.

He received rave reviews out of the gate and was performing so well by the end of his first week in the desk that guest Fareed Zakaria suggested he was staging a “brilliant, slow-motion coup.” Ratings for the long-running satirical program, which typically ebb during the summer, held steady. By August, observers had declared Oliver the heir apparent to “The Daily Show.” Instead, he landed at HBO in a deal announced in November.

The chronically self-effacing Oliver is reluctant to take credit for his successful stint, arguing the show is as strong as it is only because of Stewart’s steady leadership. He’s also the first to acknowledge he was buoyed by an unusually brisk summer news cycle. One of the year’s biggest stories, the NSA surveillance scandal, broke his very first day on the air, and was followed by a steady stream of big headlines: the birth of the royal baby, Paula Deen’s N-word controversy, Supreme Court rulings on voting rights and gay marriage and, of course, the discovery of former U.S. Rep. Anthony Weiner’s online alias – Carlos Danger.

As a host, Oliver stayed true to the tone established by Stewart while infusing the show with a boisterous, slightly goofy energy all his own. Like a true pro, Oliver even pulled off an interview with Aaron Sorkin conducted nearly in the dark after an unprecedented power outage at “The Daily Show” studio.

“I think we all felt very, very strongly that the only person who could do that job would be Oliver,” says Tim Carvell, former head writer at “The Daily Show” and now an executive producer on “Last Week Tonight.” “The only person who didn’t know it was Oliver himself.”

Though he likes to joke that, as an Englishman, he’s “not in touch with 95 percent of my feelings at any given moment,” in truth Oliver comes across as a bit of a softy. As he puts it, he cried “like a little girl” during his last “Daily Show” appearance, and his wickedly sharp humor seems to derive from genuine concern rather than condescension.

“He fully recognizes how awful and bizarre the news is,” Carvell says, “but he covers it with this weird sort of glee. There’s not a cynicism or sourness to him.”

Oliver, who for years had played the role of “Senior British Correspondent,” also discovered that he loved hosting. “After that first week,” Oliver recalls, “some guy came up to me in the street and said, ‘I’ve never seen you smile before.’ And I realized he’s probably right.”

The industry was also taking notice. Throughout the summer, Oliver lived in a self-imposed bubble, declining to take calls about possible post-”Daily Show” opportunities in order to focus on the task at hand. Just after Stewart’s return in September, Oliver spent two weeks in Afghanistan on a USO tour, well out of the reach of even the most aggressive network executive, and came home to find “a slightly bewildering amount of messages” on his phone.

By his own account, Oliver agonized over the decision to leave “The Daily Show,” where he made his debut the very day after he landed in New York in 2006. “Every part of me wanted to stay at ‘The Daily Show.’ I wasn’t really ready to go, but I don’t think I was ever going to be. I needed to be given an offer it was just insane to say no to.”

That offer eventually came from HBO, though Oliver was courted by other networks, including CBS, which met with him to discuss taking over “The Late Late Show” should Craig Ferguson depart, as well as other possibilities. HBO, home to “Real Time With Bill Maher” since 2003, had no plans to produce another politically oriented comedy show, but programming president Michael Lombardo was impressed by Oliver’s performance.

“Jon Stewart is pretty damn good at what he does. I mean this with no disrespect to him,” Lombardo says, “but by the end of that first week, that show was totally John Oliver’s.”

When Lombardo learned Oliver was finally taking calls, he and CEO Richard Plepler launched a charm offensive. “We knew that Oliver had a smart, funny, perceptive, provocative but ultimately very warm presence that we wanted on HBO.”

With a show airing at 11 ET on Sunday nights, Oliver will not be directly competing with his friends at “The Daily Show” and is insistent that even if he were, it would be “a pretty mismatched prizefight.” He also continues to turn to his mentor for advice.

“There is almost nothing I could ever do in the future I won’t be able to trace back to something Jon taught me.” Oliver says. “He is the Yoda.”

But like Luke Skywalker, Oliver has risen to become a formidable competitor in his own right. Without him to replace Stephen Colbert, who takes over for David Letterman on “Late Show” next year, Comedy Central is left with a huge space to fill at 11:30. (The network declined to comment for this article.)

HBO’s promos for “Last Week Tonight” have made light of its relative untimeliness (tag line: “Breaking News, on a weekly basis”), but Oliver thinks the slower production pace will help the show distinguish itself. With the luxury of a week to put packages together, he hopes to go in-depth on stories overlooked by the U.S. media, such as the general election currently underway in India.

“I don’t think people are going to come in and say, ‘Oh, (“The Tonight Show” host Jimmy) Fallon did this whole monologue on the Indian election.’ I think we’re all right,” he says.

Just three weeks before its debut, the show, which will tape Sunday evenings a few hours before its 11 p.m. broadcast, still existed mostly as a concept, with the exact structure still up in the air; even the set had yet to be completed.

Known for its hands-off approach to development, HBO has given Oliver, Carvell and their staff of eight writers broad latitude to create the show they want, rather than pushing them to conform to the well-worn late-night format.

“They’ve given us feedback, but it’s been so not the standard TV network advice. Their advice is all in the direction of don’t feel obligated to do the most commercial thing,” says Carvell who, like Oliver, found HBO almost disturbingly accommodating. (His theory? The network is secretly running an organ-harvesting operation.)

While leaving “The Daily Show” may not have been easy for Oliver, “Last Week Tonight” brings him to the same network as one of his comedy idols, “Veep” creator Armando Iannucci. He’s also a fan of “The Wire,” “Eastbound & Down,” and recently got hooked on “Game of Thrones,” but curbed his binge after six episodes.

“It’s a pretty spectacular pedigree of shows here,” he says. “Let’s see if we can devalue that real estate.”

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