Beautiful: The Carole King Musical, Aldwych Theatre

Stars continue to be born from Beautiful: The Carole King Musical. Following on from the ongoing Broadway run of the show, which catapulted to name status its Tony-winning leading lady Jessie Mueller, along comes the immensely likeable West End version and – oh, Carole! – much the same looks likely to happen again here.

Until this point, English actress-singer Katie Brayben was best-known for playing Princess Diana in last year's hit play King Charles III. But from this point on, and for as long as she chooses to stick with it, Brayben (pictured below) and the West End incarnation of Beautiful look set to be synonymous, the performer anchoring the slickly packaged material with total conviction and heart. The result is that, by evening's end, you actually do feel – as the song lyric puts it – as if you've got a (new) friend. 

Sincerity and a palpable lack of cynicism have always been part of the appeal of this show, which tells of the early years of Brooklynite Carol Klein, who changed her name as a teenager to Carole King and went on to pen a staggering array of songs that helped define an era. Well before she found her own voice, both literally and metaphorically along the path to female empowerment that gives this show its kick, King and ex-husband Gerry Goffin (who died last June) were writing hits for such diverse performers as The Monkees, The Drifters, and Little Eva, the last of whom was actually a babysitter in the King-Goffin household. (That realisation cues one of the funnier moments of director Marc Bruni's polished production.) 

Hers remains a back catalogue of staggering richness, and one can see its attraction to a theatre industry forever on the lookout for the latest pop-music superstar whose output is ripe for plunder, affectionate or otherwise. Where Beautiful scores and so many other trawls down memory lane have not is in the palpable affection underpinning a show that benefits from the same apparent self-deprecation exhibited by King herself. The 73-year-old icon took to the stage at the opening night curtain call to pay thanks, King's presence at the start of the London run in marked contrast to her visibility on Broadway, where the singer-songwriter took many months even to muster the courage to see the show. 

Why King's initial reluctance? She has spoken of not wanting to relive the more painful episodes included in this re-telling of her life, which makes one wonder how much Douglas McGrath's breezy but not very penetrating book has airbrushed out. We are privy, for instance, to 16-year-old school prodigy Carole's seduction by Goffin, two years her senior, with whom she went on to have two daughters even as he was drifting from one affair to another and battling bipolar disorder. In perhaps the single most dramatic improvement over the Broadway production, Alan Morrissey here invests Goffin with genuine pathos, and one feels the ache of a man who wants to do right by his family but for whatever reason simply can't. An alumnus of last year's musical flop I Can't Sing, this performer once again proves that he can in fact carry a tune – and very well. (Morrissey pictured with Brayben below.) 

Where does all this leave Carole, dramatically speaking? As a wife and mother buffeted this way and that who finally gathers the gumption to bid farewell to Goffin and her New York life and forge her career afresh as a singer as well as songwriter in Los Angeles. The framing device finds Carole having returned in triumph to the famous Carnegie Hall homecoming concert in 1971, her status by then well established: the proverbial ugly duckling has become a swan or, as another musical transplant from Broadway also dealing with an awkward young woman's coming of age might put it, she has defied gravity. 

Did the actual Carole ever act inappropriately or lose it emotionally? The show isn't saying, McGrath's book preferring multiple pointers as to how well everything is going to turn out. "I got a prediction," Gerry at one point tells her. "We're going all the way." Or as her mum (Glynis Barber, in fine comic form) says by way of response when Carole looks as if she may be hitting a brick wall: "You have your music." Elsewhere, McGrath's past work alongside Woody Allen – they co-wrote the film Bullets Over Broadway – accounts for some jauntier moments. "Vermont is too quiet; I keep thinking I'm having a stroke," or so remarks fellow songwriter Barry Mann (a strong-voiced Ian McIntosh) when he and songwriting partner-turned-wife Cynthia Weil (Lorna Want) join Carole and Gerry for a ski weekend. (The real-life Mann and Weil were also present at the press night curtain calls.)

Want is ace in the second female role – on this evidence, she'd be great with screwball comedy. A very strong cast falters only in subpar vocals from some of those playing the name artists who helped send the King-Goffin partnership on its way. (A few of the American accents need tempering, as well.) But Gary Trainor is immediately winning as Donnie Kirshner, the music producer and publisher who gave Carole a chance, and, as if a repertoire of songs that includes "So Far Away", "Will You Love Me Tomorrow", "It's Too Late", and the title number "Beautiful" weren't enough, Brayben's own modesty and charm help power the piece with much the same quiet command that, one suspects, got the show's subject where she is today. Her voice shifting with the demands of where the musical's Carole has arrived at that point in the narrative, Brayben by the end is rocking out the Aldwych Theatre as if there's no stopping her. As I suspect, once word of this performance gets around, there won't be.

• theartsdesk guide to the best musicals in London