In the much-better-than-it-sounds Amazon-backed action comedy “Heads of State,” relations are frosty between the president of the United States (John Cena) and the British prime minister (Idris Elba), until a terrorist attack forces them to team up and take out the threat themselves. There are at least half a dozen things about that sentence that might strike you as preposterous, but only one worth reiterating: “Heads of State” is infinitely more entertaining than it comes across on paper.

America has never had a president as underqualified for the job as Cena’s Will Derringer, a swollen Hollywood action hero best known for the “Water Cobra” franchise. The model for this character was clearly Arnold Schwarzenegger, whose Austrian origins preclude him from governing (but not from gubernating). Otherwise, he’s no less plausible a commando-in-chief than the matinee idols and reality TV stars who’ve done the job in this crazy, post-“Idiocracy” world.

Across the proverbial pond, Elba embodies U.K. P.M. Sam Clarke, who’s another forward-thinking choice for a country that recently elected its first person of color to the post. As written, Clarke would be the first person with military experience to serve as prime minister in nearly half a century, which makes him an actual “strongman” (so to speak), compared to Derringer, who merely played one on TV.

The surprising thing here isn’t the casting or the conception of these characters — all of which is done in the spirit of don’t-think-too-hard escapist fun — but the fact that Amazon succeeded in making such a big-screen-worthy popcorn movie. Lately, it’s starting to look like the studio is commissioning films according to some algorithm (one that privileges movies about presidents), after seeing an in-charge Viola Davis serve and protect in “G20” earlier this year and the adult sons of the president and prime minister exploring their countries’ “special relationship” in “Red, White & Royal Blue” last summer.

You could be forgiven for writing off “Heads of State.” But then, you would miss a movie that’s no sillier than “Superman” and a lot more entertaining (a comparison worth making, since both adhere to a mythical sense of heroism). After opening with a spectacular shootout involving a tertiary but important undercover character named Noel Bisset (played by Priyanka Chopra Jonas) — caught in the crossfire of Spain’s messy La Tomatina Festival — the film introduces its incredibly capable world leaders.

Newly elected Derringer is sore because Clarke had fish and chips with his opponent during the election, which the media read as an endorsement. Clarke has a hard time taking the action star seriously, since the Hollywood celebrity has zero political experience, though the prime minister’s approval ratings are precipitously low. To help patch things up, their respective advisers (Sarah Niles and Richard Coyle) suggest they share a flight to their next public appearance.

Until now, “Air Force One” — and specifically Harrison Ford growling “Get off my plane!” — represented the high point of this genre, which ranges from “White House Down” to “Independence Day” in terms of how proactively the president is allowed to defend himself. “Heads of State” leaves its endangered premiers no choice, blowing up Air Force One before the end of the first act, which finds Derringer and Clarke parachuting into a hostile corner of Belarus.

Left for dead while VP Elizabeth Kirk (Carla Gugino) deals with the ongoing terrorist threat from Viktor Gradov (Paddy Considine), the surly duo realize their chances of survival significantly increase if they work together. Let the bromance begin, as Derringer and Clarke squabble their way back to the Western world, punching out hoodlums and escaping to Poland in the back of a livestock truck.

“Nobody” director Ilya Naishuller takes gags that have no business working (like Derringer getting a face full of nanny goat teats during that border crossing) and milks them for laughs, adding original solutions to otherwise familiar action scenes (such as a mid-fight shot from inside some guy’s mouth, where the camera is positioned behind the thug’s missing tooth). Once the pair reach Poland, they meet junior agent Marty Comer (Jack Quaid), who does more creative damage in five minutes than all of “Novocaine,” and the aforementioned Bisset, with whom Clarke has romantic history.

Much of “Heads of State” amounts to overplayed action tropes, and yet screenwriter Harrison Query (who hatched the idea) and veterans Josh Appelbaum and André Nemec (who share script credit) give Naishuller and their cast ample opportunity to freshen things up, whether through original staging or reasonably funny banter. Real-world leaders would surely have better things to bicker about (where’s Armando Iannucci’s writing team when you need it?), but it’s the underlying personalities that amuse.

Petulant and insecure, Derringer is more concerned about living up to his macho screen persona than filling the shoes of former presidents, while Clarke comes off relatively erudite and confident, but sensitive when it comes to his single status. Chopra Jonas plays more than just a love interest here, holding her own in action scenes. It was smart to open the movie with her dodging bullets (and tomatoes), as that means her secret-agent credentials are well established when she resurfaces as a potential love interest.

Considine’s a great actor, but not especially menacing or memorable in the role. By contrast, Gary Oldman’s over-the-top baddie died halfway through “Air Force One,” but he still looms large in our minds. As Gradov’s security-hacking accomplice, Stephen Root makes more of an impression than his terrorist boss. The plot twists and double-crosses aren’t nearly surprising enough, nor is the soundtrack’s obvious song choices, from Gipsy Kings to the King, Elvis Presley. But in Naishuller’s hands, the action set-pieces make up for it, whether it’s the shootout in the Polish safe house or a high-speed chase in a bulletproof limo, while the variety of locations rivals a Bourne or Bond movie.

It’s been a long time since either country was led by someone who could credibly leap from a moving train or parachute out of a flaming plane (compared to America’s last couple presidents, who’ve stumbled when boarding Air Force One). Audiences may vote with other things in mind, but when it comes to take-charge action movies, fit young heroes triumph every time.