Cat on a Hot Tin Roof, Novello, London<br/>Detaining Justice, Tricycle, London<br/>Sweet Charity, Menier, London

Misogyny: check. Homophobia: check. Racism: No

Big Daddy simply is what he is, "still a Mississippi redneck", declares Maggie of her bullish father-in-law in Tennessee Williams' original Cat on a Hot Tin Roof. Actually, erase that. She can't call him a redneck in this particular West End staging of the classic family drama because – ground-breakingly – it has an all-black cast.

Here the splendid US veteran James Earl Jones (aka the voice of Darth Vader) plays Big Daddy: the plantation owner whose avaricious, bickering relatives suppress the news of his cancer diagnosis. Big Mama, his uncherished wife, is played by a comically waddling, then tearfully wounded Phylicia Rashad (from The Cosby Show). The director is Rashad's real-life sister, Debbie Allen (of Fame fame), and also joining this transfer from Broadway is Adrian Lester as Big Daddy's favourite son. That's Brick, the injured sporting hero who has become a whisky-glugging soak, hobbling around on a crutch. He is stewing over everyone's mendacity, and his wife Maggie's insinuations that there was always something "queer" about him and his beloved athletic buddy Skipper (who, we glean, has committed suicide).

Allen's African-American take may sound like a radical revision of Williams' play, but she hasn't, in fact, had to alter much of the dialogue. The legerdemain is her specified time-shift, from the 1950s to the 1980s, while retaining the setting of the grand old plantation mansion – all shuttered French windows and the spinning shadow of a ceiling fan.

The playwright himself kept on redrafting this piece as late as 1974, having been frustrated by the initial, toned-down stage and screen versions of it. Now, drawing on the '74 script, this production clearly foregrounds Maggie's accusation of homosexuality, which Brick furiously denies. Jones's mouthy bluntness is startling and explosively funny too, as his Big Daddy hurls the F-word around, trashing the superficial niceties of his 65th-birthday party.

I wonder if Allen mightn't have been braver still and updated the action to right now. The issue of gay sportsmen fearing to emerge from the closet remains topical enough. (Take the basketball pro John Amaechi coming out in 2007, only to face an anti-gay rant from his fellow player, Tim Hardaway.)

The sort of unreconstructed, dismissive misogyny displayed here – by both Big Daddy and Brick towards their wives – has hardly been consigned to history either. It remains, alas, instantly recognisable. What's ultimately striking is the race- and decade-straddling relevance of this dysfunctional domestic drama.

It has to be said, Williams' original whites might have looked more like a doomed dynasty. Sanaa Lathan's slinky Maggie doesn't fully convey the feverish insecurity underlying her relentless yakking when alone with her stonewalling spouse. Allen's directing has a few naff touches as well: a crass spotlight for Rashad's "big speech" about life's brevity; and fractionally overplayed comedy.

Lester, however, is superb, with hatred simmering beneath his cold, depressed silences. He manages to be funny and ferocious when he explodes, hurling his crutch at Maggie like a thunderbolt. Meanwhile, in the great father-son confrontation at the heart of the play, Jones is richly complicated: slumped like a sour-mouthed bulldog, barking rebukes, but then suddenly beaming with paternal love and hidden depths of tolerance. Even when he garbles his lines a bit, Jones turns it into rambling naturalism, and his intimate physical tenderness is unforgettable – momentarily cradling his son's head, or wearily lolling on Brick's shoulder before he turns and snarls again. Timelessly poignant.

Rounding off the Tricycle's Not Black and White season, Detaining Justice by Bola Agbaje zooms in on immigrants in contemporary Britain. A brother and sister ironically called Justice and Grace (sharply played by Aml Ameen and Sharon Duncan-Brewster) have each fled Zimbabwe. Though Justice, the brother, suffered persecution there, having opposed Mugabe, it is Grace who arrived in England first and has been granted asylum. Trying to slip in undetected, Justice is now bitterly festering in a detention centre, not believing his timid sister is pleading with the Home Office, and degrading herself to get him out. Though this central plot-strand is under-developed, Agbaje is a bold and promising writer, remarkable for her ability to combine bleak experiences with comic relief.

Finally, the Menier has another hit musical on its hands with Sweet Charity, Neil Simon's 1960s Manhattan romcom, with jazzy score by Cy Coleman. Tamzin Outhwaite is the titular heroine, a hostess in a sleazy dance hall who never tires of hoping to meet a dreamboat. Too bad Outhwaite's superglued showbiz grin and nasal American accent lacks genuine charm. And the live band, at full volume, is so brassy it hurts. And yet, crammed into this snug venue, director Matthew White's troupe are having a joyful blast, and that's infectious.

Mark Umbers is absolutely splendid, multi-tasking as Charity's string of sweethearts, morphing from the moustachioed smoothie Vittorio, into the hilariously neurotic Oscar, working himself up into a hyperventilating, squeaking frenzy when stuck in a lift. Outhwaite is also very droll when jammed in Vittorio's (translucent gauze) wardrobe, chomping on a ham roll and mouthing in synch to his mistress's histrionics.

Best of all, the show-stopping numbers are thrillingly good, choreographed by Stephen Mear. "Big Spender" is chorused by the jaded hostesses with tarty voguing slowed down to chilling, dead-behind-the-eyes robotics, and they fade to a whisper like a cynical sigh. Another West End transfer is surely in the pipeline.

'Cat on a Hot Tin Roof' (0844 482 5170) to 10 Apr; 'Detaining Justice' (020-7328 1000) to 15 Dec; 'Sweet Charity' (020-7907 7060) to 7 Mar