“The Courier” is an engrossing espionage thriller set, as so many of them are, at that one point when the Cold War seemed most likely to turn nuclear hot.

Well-acted and early ’60s period perfect, about the worst you can say about it is that it’s a washed-out copy of “Bridge of Spies,” which is its easy analog and far more suspenseful and immersive.

Benedict Cumberbatch is our unlikeliest of “heroes,” a chain-smoking, elbow-bending British salesman coerced into carrying messages between a Soviet turncoat and the West in the blustering, impulsive last years of Nikita Khrushchev’s rule.

How MI6, at CIA urging, settled on Greville Wynne as their go-between is skimmed over in Tom O’Connor’s script. A chance meeting that an espionage operator (Angus Wright) recalls in a pinch, a lunch date arranged, with a CIA intermediary (Rachel Brosnahan), fine food and a bit of booze and flattery and appeals to patriotism seal the deal.

Wynne’s bonafides are that he represents British firms wanting to sell machine parts, and has that salesman’s gift of remembering names and knowing when to lose when golfing with a client. He likes to eat, drink and smoke and travels the Eastern Bloc on sales calls. In “The War” he was a mere private who “never saw action.”

“The truth is, if this mission was dangerous, you’re the very last person we’d send,” his handler-to-be (Wright) purrs.

The Whitehall use of “mission” in an Oxonian accent should scare Wynne off. But he reads the papers. He might prefer to make sales inroads to the USSR “when things are a mite cooler,” tensions-wise. His would-be handlers’ leverage is simply this. Nuclear war is edging closer and the CIA has no inside information about Kremlin thinking or planning. If this Russian (Merab Ninidze) who has reached-out can’t help, the world’s only going to get more dangerous. Wynne relents.

Veteran stage director Dominic Cooke and screenwriter O’Connor skimp on the “tradecraft” — mere suggestions that “every Soviet is the eyes of the State,” a package hand-off here and there — building their thriller on the personal relationship between the salesman and the spy-trained Soviet who is a higher up with a ministry in charge of industry and trade.

Oleg Penkovsky is a father fretting over how “unstable” Khrushchev seems to him, and is desperate to “prevent a war.” He is also a spy, part of whose job “is to steal technology from the West.” He is confident and reassuring to his inexperienced “courier” of documents and microfilm.

“I am better at this than ‘they’ (the KGB/GRU) are.”

But hanging their film on that partnership depends on chemistry and the sense that a deeper bond is forming, as “Alex,” as he prefers to be called in English, escorts Wynne around Moscow, visits Britain on a trade trip and meets and dines with Wynne’s family (Jessie Buckley, Keir Hills).

We get plenty of the wining, vodka-ing, smoking and dining, and in the film’s cleverest touch, proof of Wynne’s grinning answer to that most important query if you’re trying to fit in with and trick Russians.

“Can you hold your alcohol?”

“It’s my one true gift!”

It’s that extra pathos, that intimacy that would personalize the stakes in this movie about a world lurching toward the Cuban Missile Crisis, that’s sorely missed here.

Cumberbatch makes Wynne a Brit of his times — reserved, stoic, suppressing emotions as he lives this secret life that his suspicious wife doesn’t know about. We get a hint that the salesman’s soul is finally reached when he learns to appreciate Russian ballet (“Swan Lake”), but little else that explains his later actions. The picture stumbles into a dry-eyed third act that should be wrenching but plays here as an extended epilogue.

Cumberbatch’s commitment to the role is most impressive in that stumbling third act, but this internalized performance is more historically appropriate than audience-endearing.

Buckley and Brosnahan are the players who give us more to latch onto than our buttoned-down leads. And if we’re handing out plaudits, kudos on landing a wonderful big screen Khruschev (Vladimir Chuprikov), and keeping the Soviet scenes “Russian” (with subtitles).

Our KGB villain (Kirill Pirogov) may not have enough scenes to make a big impression. But “The Courier” does a passable job of passing on the paranoia of a country where betrayal and summary arrest could come from anyone, at any time.

It’s that “passable job” that’s the rub here. As this “true story” hews closely to the plot points of many a spy thriller, “The Courier” invites comparisons that highlight its shortcomings, telegraphing punches that we sense are coming and failing to ever land a telling blow.

MPA Rating: PG-13 for violence, partial nudity, brief strong language, and smoking throughout

Cast: Benedict Cumberbatch, Merab Ninidze, Jessie Buckley, Angus Wright, and Rachel Brosnahan.

Credits: Directed by Dominic Cooke, script by Tom O’Connor. A Lionsgate film.

Running time: 1:51