Ogden Nash wrote that “Mr J. B. Priestley is simply beastly”, no doubt reinforcing the dramatist’s belief that New York, where his plays regularly flopped, was inhabited by “the silliest people the world has ever seen, entirely without sense or roots”. And in the prePeter Cook era, the Cambridge Footlights staged a sketch showing J. B. Beastly watching a play in which a Chorlton plumber and his clippie wife say such things as “Aye, home’s best” and “All that glitters is not gold”, while he cried “author, author” from a box. For another southern snob, Virginia Woolf, the doughty Yorkshireman was “a tradesman writer”.

Priestley was the son of a Bradford teacher, left school in 1910 at 16, but, after enduring horrors galore in the First World War, got an MA from a Cambridge that he thought packed with shallow, spoilt men. They, Bloomsbury and America found him solemn and pontificating, which he could be. But they missed his importance. As a national tour of Stephen Daldry’s great production of An Inspector Calls is now showing, and Rupert Goold’s impending revival of Time and the Conways may confirm when it opens tomorrow, he was the playwright who kept serious drama alive during the frivolous 1930s and 1940s.

His foes also missed his sense of humour. Personally, he could be grim and depressed — some thought because he had been traumatised by a war that had “sliced my generation into sausage meat and held it above a swill bucket” — but also warm, funny and the jaunty womaniser who counted Peggy Ashcroft among his lovers. When We Are Married, now playing at the Everyman in Liverpool, is a terrific comedy: a genial, mischievous portrait of the respectable northern bigwigs who discover they were never properly wed.

Whether he was writing tragedies, comedies or a mix of both, Priestley was scathing about hypocrisy, pomposity, callousness, selfishness, cynicism, idleness and avarice. When the title character of An Inspector Calls accuses a rich, smug family of complicity in the suicide of a poor girl, his targets are as universal as those of a medieval morality play, even if they’re transposed to the northern town he variously called Brumley and Burnanley. But when he’s attacking what he saw as the neurotically obsessive greed of money-men interested only in making yet more money, as he did again and again, he seems topical too: which may be why he’s so firmly back on the theatrical map.

It would be nice to see a revival of Priestley’s cheerfully scurrilous Laburnum Grove, in which a boring suburbanite is revealed as a master forger, who mildly explains that he’s helping the country through economic depression by remedying the “unhappy fact there isn’t enough money in circulation”. Change his name from Redfern to Brown, and he’d be a new Labour hero. But there are also Priestley plays that date. Who would revive They Came to a City, in which characters visit a Utopia where all is co-operation and sharing — some hating it, some loving it and staying, but the best of them returning to Earth to transform our world into that city?

That reflected not only the idealistic socialism that Priestley inherited from his Baptist father but the optimism of 1943, when people were beginning to expect victory and social change. He was always responsive to the concerns of his time, which is why the mood of The Linden Tree, the play that followed in austere 1947, is very different. If one character is a “deluxe modern spiv”, making merry before the A-bomb destroys a Britain that’s “like a dingy funfair”, another is a woman commissar, a control freak who wants labour camps for wastrels and dissidents — and embodies the cold socialism that Priestley loathed.

Time is always significant in his plays. Several are set in Edwardian Britain, an era he regarded with nostalgia and Cassandra-like dismay. Progress and science mean there’ll be no war, say complacent characters in Eden End and An Inspector Calls, the wealthy industrialist in Inspector adding: “Why, look at the new liner, the Titanic, absolutely unsinkable!” Indeed, Priestley actually wrote three “time plays”, of which Time and the Conways is the best. Its portrait of a family in 1919, in 1937 and again in 1919 was inspired by the now-forgotten theories of J. W. Dunne; but it’s more about the fragility of hope, the sadness we’d probably all feel if we discovered that we knew what lay ahead.

Priestley wasn’t a subtle writer. His characters are often attitudes, sometimes even caricatures. He ruefully mocked his own dialogue as “that familiar flat idiom”. Yet his work is impressively various. There’s the thriller Dangerous Corner and the “tragic farce” Bees on a Boat Deck, in which representatives of big business, communism, fascism, pure science and hedonism assemble on a doomed ship symbolising England.

And there’s Johnson Over Jordan, with Priestley’s favourite actor, Ralph Richardson, playing a small-time Everyman who joins Mr Rat and Mr Slug in an expressionistic nightclub, picks up his own daughter, kills his son — but is redeemed after learning the lesson that Arthur Miller preached in a more famous play: they’re all our sons, all our daughters.

Priestley was sometimes accused of denouncing flaws, sins, crimes but never acknowledging the depth of human evil. Yet listen to the surprisingly sympathetic Cabinet minister in his odd, experimental Music at Night. Since 1914, he says, “we’ve all existed in a series of vast madhouses, shrieking with hate and violence, stinking of death”, and he goes on to compare himself to the paralysed, half-blind driver of a huge, fast bus whose passengers are reading the cricket scores. It’s yesterday’s voice, but, like Priestley’s own, it has something to tell today and tomorrow.

Time and the Conways is at the Lyttelton, SE1 (020-7452 3000; www. nationaltheatre.org.uk), to July 27. When We Are Married is at the Everyman, Liverpool (0151-709 4776; www.everymanplayhouse.com), to May 23

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