James Bond was the daddy, obviously, but the Cold War airwaves were full of spies. There were the serious, broken-hearted spies created by John le Carre, spoof spies such as Maxwell Smart and – by far the largest category – glamour action heroes such as the stars of the popular television series The Man from U.N.C.L.E. That was Bond with a twist, one might say, although the biggest twist was the fact that the enemy Russian spy turned out to be everybody's favourite.

The Man from U.N.C.L.E.'s big idea was that the United States and their Soviet enemies are forced to work together in secret to defeat THRUSH, an unspeakably evil organisation dedicated to taking over the world. Robert Vaughan played Napoleon Solo, the American ladykiller in a Savile Row suit; David McCallum, who was actually English, played Illya Kuryakin, who wore Beatnik roll-necks and suede jackets and was moody and mysterious. No prizes for guessing whose picture adorned the walls of teenage girls across the Free World.

Clearly, The Man from U.N.C.L.E. was a concept just waiting to be dusted off by somebody. Guy Ritchie, who has had such a success refurbishing Sherlock Holmes, took this project over from Steven Soderbergh, who walked away after years mired in discussions with the studio. Ritchie ditched Soderbergh's script – "It wasn't a movie I would have seen" – and set about getting the bones of the story in place.

We meet on set when he is shooting a studio set as 1960s East Berlin; it is midnight in the dead of winter, but he is cheerful enough.  "Once you think the narrative through, the embellishments can start," he says, "which is where I really start directing."

His original plan was to cast Tom Cruise as Napoleon Solo and play him off against a much younger Kuryakin, but scheduling clashes made that impossible. Henry Cavill, the new Superman, was cast instead; Armie Hammer plays the Russian. The two of them come over to talk while the crew prepare their zipwires; they are about to scoot over a fence to the top of a scaffold on harnesses, which doesn't seem to faze either of them.

Cavill says that he's been learning not to take himself too seriously. "Someone who's just too cool and never gets anything wrong is less likeable than someone who makes little mistakes, who makes kind of a fool of himself every now and then. I learnt that."

In the original series, the characters had negligible backstories; all you really needed to know was that Kuryakin quoted a lot of poetry. In this version he is, according to Hammer, "a born and bred spy; there's nothing he enjoys more than being a KGB operative". It is Solo who is the square peg, an art thief who has been blackmailed into working for the CIA. "He has a skill set the CIA can use," says Cavill. "But he's reluctant to use it." And the original series certainly didn't feature any women warriors.

Alicia Vikander plays Gaby, a mechanic who becomes embroiled with the international spying duo; they soon get her out of her overalls and into a succession of swinging '60s op-art shifts as she poses as Kuryakin's trophy wife while driving the getaway cars. The first thing she had to tell Ritchie, she laughs, was that she can't drive. Even pretending to drive in a car of the period was hard enough. "I had a two-hour lesson where all I did was make U-turns and 180s. I was so sore because it was so heavy."

Not that she was alone: Australian actress Elizabeth Debicki, who plays an exceptionally well-groomed villain, can't drive either. Debicki is not on set, but we are able to visit her criminal HQ, which has a very '60s conversation pit in the middle of the room. We also get to press knobs on the bridge of a British Navy aircraft carrier and see the "marbled" couture boutique where Gaby is kitted out.

It's all very opulent, but playful, too.The fact is, Ritchie says in an interview I read later, real spies would be bureaucrats. "But I like that, you know. I like the idea of taking essentially what's a sort of boring, officious job and turning it into something that is a fantasy to a degree. I like making that entertaining."

The Man From U.N.C.L.E. opens on August 13.