Another opening of another show. The Dresser, Ronald Harwood's 1980 play about an actor-manager in the 1940s and his dresser, has had some notable casting: Albert Finney and Tom Courtenay in the 1983 film and, just last year, Ian McKellen and Anthony Hopkins in a TV version.

But Ken Stott as Sir and Reece Shearsmith as his dresser Norman make the parts their own. Under Sean Foley's direction, this is an altogether brighter and broader backstage comedy than that recent prestige-TV outing. It's played for laughs, and laughs it very much brings – although such an approach also amplifies the creakiness of Harwood's play. The Dresser may be sending up a bygone era, but 36 years on, the play itself feels almost as dated.

Sir ought to be getting ready to give his Lear, but he's in the midst of a break-down, weeping copiously. Whether this is due to age and exhaustion, or whether he's suffering mental health issues, isn't clear. But the show must go on - according to these two tragic co-dependents, at any rate - and only Norman, his prattling fool, knows how to manage him.

The show must also go on despite the damn Nazis: that's right, there's an air raid just as the curtain goes up. But the Germans will never take away our Shakespeare! To be fair, Foley's production retains a degree of ambivalence in the face of potential gross patriotism – Sir's rather absurd personal fury with Herr Hitler punctures the luvvie notion that standing on stage is the height of bravery.

Indeed, by going for all-out comedy, this production is pretty unsentimental – we don't really see Sir as a towering talent, and it's a little hard to understand why everyone's so fawning. He has a noble dedication to his art, but it's really because he's monstrously self-regarding and an emotionally blood-sucking egomaniac.

Here, he's also a comically past-it figure, with slapstick scenes where he fails to put on his trousers or paints his face with the wrong pan stick ("Not Othello!"). Stott has a twinkle, and mines Sir's confusion (dementia?) for both laughs and sadness, but it never gets close to Lear-esque levels of tragedy.

Shearsmith is terrific as Norman. A wheedling, waspish presence, full of tattling tales and bitchy one-liners, he's extremely funny. It's a physically comic performance too: his tottering trot, lip-pursing and limp-wristed gesturing are all over-the-top, but it works – as if this thwarted thesp has accidentally absorbed a little of the grand style from the wings. But the pep is also cut with possessive menace; Norman suddenly seems like a character from one of Shearsmith's own black comedies, Inside No9 or The League of Gentlemen. It's a treat.

Elsewhere, the gags groan like Lear's death scene. That Sir's partner (the magnificently regal Harriet Thorpe) is a bit heavy to carry on as the dead Cordelia is funny once; as a repeated trope, leading to a scene where Sir gropes a young actress to see if she'd make a skinnier replacement, it is waring. That such a scene is played with a cutesy comic coyness, rather than anyone onstage looking remotely bothered, is another sign that Foley hasn't quite worked out how to stage this play for today.

The Dresser runs at the Duke of York's Theatre until 14 January 2017.