Phèdre, NT Lyttelton, LondonThe Winter's Tale/ The Cherry Orchard, Old Vic, LondonKursk, Maria, Young Vic, London

Helen Mirren fails to rise to the challenge of 'Phèdre' at the National, while Sam Mendes's Old Vic twin project is a classical treat

Helen Mirren's doomed Queen emerges from her palace, her face veiled in purple silk. This troubled wife of King Theseus is concealing a dark secret in Racine's neoclassical tragedy, Phèdre. As soon as she steps gingerly into the loggia – a monumental chasm carved out of a cliff in Nicholas Hytner's production – Mirren collapses in a heap, and blubs like a hysterical girl.

Eventually, the hideous truth comes out. The queen is infatuated with her stepson, Hippolytus (Dominic Cooper). In turn, the prince is hopelessly enamoured of Aricia, an enemy-princess imprisoned by his irascible father. It's a fatal tangle of forbidden desires, a tragedy exacted by punitive gods. As personal disasters go, this one's off the scale.

But the real disaster is this: you don't give a monkey's. Neither Mirren nor Cooper convey any great emotional depth, either of searing visceral passion or repulsion. This is particularly disappointing and dull after Cheek by Jowl's recent sizzling production of Racine's Andromaque.

Hytner's staging looks gorgeous but that, alas, isn't enough when his leading players don't grapple properly with the verse – a vigorous adaptation by Ted Hughes. Mirren superficially goes through the motions of anguish and ardour while Cooper (known to many from The History Boys) is inclined to declaim his lines in great slabs. Let's hope they hit their stride by 21 June, when the NT Live scheme is due to be launched, with this Phèdre simultaneously screened in 170 cinemas around the world.

At present, the show's true star is John Shrapnel, playing the prince's quietly loyal confidant, Théramène. His climactic speech – reporting how a tidal wave and furious sea monster have liquidated Hippolytus – is fabulously gripping, superbly paced. It brings a surge of brilliance at the close.

In The Winter's Tale, King Leontes' fit of jealousy devastates his family because he sees an illicit love triangle where there is none. Imagining that his wife, Hermione, and his old friend, Polixenes, are adulterers, Leontes turns lethally destructive and ends up losing everything he has held most dear – including his princeling-son Mamillius.

Of course, Shakespeare's late romance doesn't end there. It moves beyond tragedy, through the passing years, to forgiveness and love resurrected. Thus the great scheme of things – in a profoundly consoling form – is embedded in Sam Mendes' ambitious new Bridge Project.

Pairing The Winter's Tale and Chehkov's The Cherry Orchard as its first double bill, this UK-US enterprise will present classics annually, co-produced by the Old Vic, the prestigious New York venue BAM, and Mendes' own Neal Street Productions.

In The Winter's Tale, Hermione's show trial is electrifying and heart-rending. Rebecca Hall proves herself a great young classical actress: beautiful, willowy, nearly broken with grief, yet rising to fiery anger. Simon Russell Beale, most strikingly, suggests his insecure Leontes knows, deep down, that his wild accusations are false. He perversely clings to them, though, like a stubborn child.

Mendes also makes the trial scene seem poignantly like a furious domestic row, by paring down the courtroom to a long pine table, with husband and wife sitting at either end.

Yet Mendes' touch isn't completely sure. There's a surprisingly saccharine and gimmicky moment right at the start where Mamillius (the American actress Morven Christie, who doubles as Perdita) talks to a teddy bear by way of a prologue. When we are transported to pastoral-comical Bohemia, a hoedown with phallic balloons goes down like a lead one. And the mainly Edwardian costumes aren't made to seem especially apt, being principally there to persuade us that the two plays are natural twins – which they ain't.

Sinead Cusack is not on top form, either, in The Cherry Orchard. As Ranevskaya, the sentimental landowner who loses her estate to Beale's peasant-born Lopakhin, she does far too much mournful gazing out, over the audience. Still, Mendes trenchantly foregrounds Chekhov's awareness of the coming Russian Revolution. The lone beggar, who passes through, symbolically grows into a long silhouetted line of paupers, like a massing army.

Some of the actors' appearances in both plays create engaging echoes too, with little continuities and variations. Beale's socially clumsy Lopakhin and Hall's shy Varya are touching as the Chekhovian couple who never get together. Richard Easton's twinkly silver-haired incarnation of Time becomes Ranevskaya's terminally aged servant, Firs, in funereal black. Meanwhile, Ethan Hawke transforms winningly from Shakespeare's vagabond Autolycus – a grinning pickpocket – into Chekhov's radical intellectual Trofimov who wanders away into the future, waving away banknotes, hoping for a better world.

Finally, we come to Kursk. Here Russia has moved on into the post-Soviet era at a rate of knots. The troupe Sound&Fury has turned a studio theatre into a crowded submarine for this semi-promenade piece which imagines what went on in the Barents Sea in 2000 when an explosion sank the titular Russian sub.

The low-tech set design is pleasing: a dimly lit space from which glinting ladders ascend to the bridge. Bunk beds, showers, and the captain's cabin are crammed against the surrounding walls, and the control room squats in the centre. The mechanics look rather like Meccano, but the sound effects of muffled propellers and sonar beeps are eerie, as is the plunge into frightening, quietly dripping darkness after the explosion.

Only we're not actually on the doomed Kursk. We're on a fictional, snooping Royal Navy sub nearby, being served up an unconvincing script by Bryony Lavery, uneven acting, precious little illuminating info, and some misleading dialogue about the British offering no rescue assistance. Hey ho.

'Phèdre' (020-7452 3000) to 2 Aug; 'The Cherry Orchard' and 'The Winter's Tale' (0844 871 7628) in rep to 15 Aug; 'Kursk' (020-7922 2922) to 27 Jun