ITV’s new version of The Ipcress File began with a close-up of a pair of black-rimmed glasses just like those worn by Michael Caine in the 1965 film. They were then put on by their owner (Joe Cole), thus transforming him into Harry Palmer – but also neatly establishing the kind of show we were in for.

Sunday’s first episode did a fine job of setting up an impeccably twisty (i.e. confusing) Cold War plot. It spared no effort in its quest to show us that the Britain of 1963 was on the Brink of Social Change. And yet, neither of these things really got in the way of its main aim: to be as cool and stylish as the 1960s films and TV it was paying tribute to. Happily, not only did it succeed, but it also pulled off their fun trick of hovering around the borders of silliness without ever crossing them.

In that opening scene, Harry (together with the naked hottie his specs revealed to be occupying his bathtub) was in West Berlin, where he was serving as a soldier. But not, it turned out, for long. Arrested for an impressive range of black-market activities – including supplying fresh lobster to the East German mistresses of Russian generals – he was taken back to Britain and banged up in prison. But not, it turned out, for long.

That’s because in the meantime a leading British nuclear scientist (his boffin status duly signified by the wearing of a bowtie) had been kidnapped and the chief suspect was one of Harry’s old black-market contacts. As a result, he was visited by Dalby (Tom Hollander), head of a bespoke government spying unit, and made an offer he couldn’t refuse. He was to go to Berlin, contact his contact and bring the boffin back.

But first, of course, Harry and Dalby were required to exchange some frankly arch dialogue over lunch in a swanky London restaurant. By the time the bill arrived, we were fully up to speed on Harry’s backstory, with particular reference to being a representative of the new socially mobile working class.

Not that the brink-of-change stuff ended there. Another of Harry’s handlers is Jean Courtney (Lucy Boynton), an archetypal 1960s glamour-puss as regards clothes, hats and hairdos – but not when it comes to fulfilling her posh parents’ expectations that she should marry her posh fiancé, abandon what they think is her BBC career and settle down to a life of jewellery-wearing. Instead, she headed for the American embassy to gain safe access for Harry in Berlin, and to be startled that her opposite number wasn’t white. ‘Yes, that’s right, I’m black,’ he admitted cheerfully – adding, perhaps anachronistically, that ‘It’s all good’.

Except that, before long, it really wasn’t. Harry did track down his contact, who in an all-purpose foreign-bad-guy accent greeted him with the words ‘Nobody could get lobsters like you, Harry, but if I think you’re trying to trick me…’ – amplifying the second point by holding a knife to Harry’s throat. Yet, even after Harry seemed to have convinced him otherwise, the boffin-reclamation scheme went spectacularly and intriguingly wrong.

What’ll happen next – or indeed what’s happened so far – is anybody’s guess. (Although I think we can confidently predict that everyone involved will continue to speak and act not like people in real life, so much as characters in cool and stylish spy films.) But after Sunday’s episode, it looks likely that we’ll have a blast finding out.

So, what if Alan Partridge was a hunter of witches in 17th-century England? This question appears to have been the starting point for The Witchfinder, written by two of Steve Coogan’s regular collaborators on Partridge projects, and starring Tim Key, one of his sidekicks in This Time with Alan Partridge. Fortunately, if the question is somewhat unexpected, the answer seems to be that we’d get a hugely promising new sitcom.

Key is terrific as Gideon Bannister, a man whose bumptiousness is matched only by his lack of self-knowledge – although, like Alan, somewhere deep inside is a hideous and not entirely suppressed awareness that he’s a bit rubbish. Nor, try though he might, can he quite mistake his career for a success, as he pitches unavailingly for serious witchery cases instead of the odd dead pig he ends up with. Worse, the latest woman he’s accusing (Daisy May Cooper) makes few bones about pointing out what a pillock he is.

In this case, then, the borders of silliness are not just crossed, but trampled with infectious glee. On the face of it, the desire to see women burn is not an obvious comedy subject. Here, however, it’s treated with such wild, knockabout, reality-avoiding abandon that it’s hard to work up much indignation – or much guilt – when line after line proves properly funny.