An actress on tour with “Hamlet” is “recruited” by Britain’s most notorious spy in 1950s Moscow in “An Englishman Abroad,” a delightfully droll tragi-comedy from the writer who gave us “The Madness of King George,” “The Lady in the Van” and this year’s “The Choral.”

It’s a true story, or true enough. Posh, upper crust, Cambridge gay blade turned British Intelligence officer Guy Burgess fled to Moscow upon being outed. And a drunken, exiled Burgess went out of his way to meet actress Coral Browne when the Old Vic’s latest “Hamlet” reached the U.S.S.R., as totalitarian Red Russia branded itself back then.

Put that anecdote in the ear of one of the British theater, TV and film’s greatest modern wits — Alan Bennett — and it became a John Schlesinger (“Midnight Cowboy”) TV movie for the ages starring Alan Bates as Burgess, Browne (“The Ruling Class,” “The Night of the Generals”) as herself and Bond-villain/”Rocky Horror” “criminologist” and narrator Charles Gray.

Burgess was the most famous member of the “Cambridge Five” spy ring who infiltrated British intelligence during World War II and served the Soviets until they were exposed in the early years of the Cold War. He defected in 1951.

We meet him drunk and dozing off at a touring “good will” British production of “Hamlet” in Moscow in 1958. He barges into Browne’s dressing room during intermission. She was playing Queen Gertrude opposite Gray’s usurper/poisoner King Claudius, an actor Burgess tipsily badgered several stern Soviet female theater functionaries to see backstage.

“We were at Cambridge together!”

The jokes begin with his muttering about at long last hearing a better language than Russian and the most eloquent words ever written in English to other patrons as he stumbles out of the theatre. Was he “friends” with Hamlet or Claudius? Both, it turns out. “I was at Cambridge with Hamlet!” he later insists out of vanity.

Hamlet, by the way, is played by Mark Wing-Davey of “Hitchhiker’s Guide to the Galaxy” fame. In real life, the theater troupe’s Hamlet was Michael Redgrave, also a college friend of Burgess.

Burgess throws up in Browne’s dressing room sink, charms her with his boorishness, cadges booze and English smokes.

“Craven A, for your throat’s sake,” he sighs with pleasure at the memory of finer tobacco.

Browne is bowled over, not sure who he is or why there’s a young man tailing him whom she catches listening at a keyhole.

“I’m in a French farce!”

But after the curtain call, she confers with her Claudius (who is never mentioned by his real name, Mark Dignam), who confirms who Burgess was and that he did indeed know Burgess in university as they traveled in the same (theatrical and gay) circles.

“Oh I used to run across him…years ago..the way one does.”

It turns out Burgess isn’t done with Browne. He sends her a note inviting her to lunch. It takes a day of walking, questioning Russians who don’t understand her for the address and swapping insults with a couple of young, snide foreign service officers at the British Embassy before they finally meet.

What has been a “How DOES one get street directions in a paranoid totalitarian state where you don’t speak the language” farce — “like playing ‘Private Lives’ to a Wednesday matinee in Oldham!” — tranforms into a deliciously downbeat variation of the Portrait of a Broken Man cliche.

Bates, one of the great actors of his generation, was never better than he was as Burgess putting up a front that he has no regrets — save for not getting new English dentures before fleeing — and who only misses the “gossip” of Olde London Towne. “Treason” is a word he won’t hear of. Only Browne — a native Australian — has the nerve to bring it up.

The homosexual business? That is and never was a secret, Burgess insists, along with his communist sympathies. He lets on that he’s living his best life, even if his messy, depressing flat’s live-in lover may have been “assigned” as part of his team of minders.

Speaking Russian? “I should learn it, if only for the sex!”

Burgess sings Church of England hymns in the toilet and Gilbert and Sullivan with his balalaika-playing lover (Alexei Jawdokimov) quotes Shakespeare and “Tristram Shandy” from memory and carries on as a shabby bon vivant keeping up a brave front.

He did request that his new actress friend bring her tape measure, after all.

Browne, playing a self-effacing/self-mythologizing version of herself in all this, is both patriot and pragmatist, an apolotical “actress” actress who pities the fallen man and tells him as much.

Her “no one will ever believe me” about all this when she gets home is the best reason to not go to the British newspapers. Kindness is another.

Schlesinger, famed for “Midnight Cowboy” and “Marathon Man,” set himself up for British romps like “Cold Comfort Farm” and an early British TV version of “Sweeney Todd” with this darkly amusing riff on the corrosive corruption of class — the reason Burgess got away with all he did for so very long.

Schlesinger and Bates made four films together — “A Kind of Loving,” “Far from the Madding Crowd,” “Sunday Bloody Sunday” and this, one of the most critically acclaimed of the lot. Browne and Gray and assorted bit players shine in roles supporting the fatal flaws of “class consciousness” messaging — tailors, cordwainers, etc.

Bennett’s dialogue is so delicious that one is tempted to stop and replay savory lines.

“Our friends, the foe, are just beginning to ‘play ball’,” the insufferable, sexist foreign service swells lecture Browne. They can’t have her upset the early version of “detante” that Kruschev ushered in before The Berlin Wall and the Cuban Missile Crisis sealed his fate and reputation.

The grand scale of Tsarist “imperial” locations as well as impersonal, high-rise “Soviet” seeming apartment blocks in the drabness of winter was recreated by filming in Dundee, Scotland.

The only twist that might have made this grey, quippy “Spy Who Came in from the Cold” era dramedy better would have been televising it in Orwell’s fated year 1984, but then one can’t have everything.

This seldom-available perfect jewel of a 60 minute telefilm is now on Amazon, so see it while you can.

Rating: 16+, profanity, innuendo

Cast: Alan Bates, Coral Browne, Charles Gray, Roger Hammond, Douglas Reith, Peter Chelsom and Mark Wing-Davey.

Credits: Directed by John Schlesinger, scripted by Alan Bennett. A BBC film on Amazon Prime.

Running time: 1:01