How cheering it is to see this beautiful, comic fantasy by Bernard Shaw, in which poor little Cockney flower-girl Eliza Doolittle (elegant Michelle Dockery) vaults class barriers and achieves upward social mobility, thanks to elocution lessons with Professor Higgins. Peter Hall’s thought-provoking production, admired at Bath last summer, made me think how Cherie Blair relates to Pygmalion’s 1912 world. Sir Peter offers no such linkage, but Mrs Blair and Eliza Doolittle are spook-ily alike, although Eliza displays the good taste and style to which Mrs Blair and her new autobiography are strangers.

The fictitious and real-life women were each born to poverty and saddled with unreliable fathers. Cherie, however, was born in a period when educational opportunities available to all classes and innate intelligence enabled her to class-climb and fulfil potential, becoming grand enough to be snubbed by Princess Anne. Shaw’s politically pointed play, which makes delicious fun of the rigidities of the Edwardian class system, serves a poignant reminder, therefore, of social and cultural change for the slightly better. Dockery’s doleful Eliza, a girl with a daddy complex caused by the neglectful Doolittle (Tony Hay-garth), is left vainly hankering for the sexually ineffectual or sexless Higgins, and the hope of teaching other Cockney girls how to rise socially.

Sir Peter opts for an uncharacteristically traditional production, with longish pauses between acts, after Simon Higlett’s handsome designs for porticoed Covent Garden vanish and Higgins’s wood-panelled, leathery sitting room and his mother’s Chelsea drawing room are assembled. The comedy of bad manners, which drives the inter-changes between Higgins and Eliza, comes forcefully across. Tim Piggott-Smith, in dynamic form, portrays the Professor of Phonetics as a dishevelled bundle of nervous energy and theatrical extroversion. Sprawling, restless and loose-limbed, eccentric in his childlike self-centredness, he treats Dock-ery’s spirited but not that Cockney Eliza as if she were some intriguing laboratory specimen.

The scene of the girl’s first social outing, an at home given by the professor’s mother (Barbara Jef-ford) becomes the occasion for precise though not very forceful mockery of middle-class manners. It parades Higgins in bored, social blunder mode while Dockery’s coolly transformed Eliza, in eyecatching white, behaves too much like a well-spoken mechanical doll to carry total conviction, though the first night audience adored the girl’s innocent faux-pas right down to the triumphal “Not bloody likely.” Jefford, a great, scandalously under-used classic actress, provides an unmissable, matchless display of maternal exasperation and social civility as Higgins’s mother, expressed in the perfectly modulated tones of a humanised, more liberated Lady Bracknell.

Piggott-Smith interestingly deals with the unresolved nature of Higgins’s feelings for Dockery’s disconsolate but not desperate Eliza by rendering him emotionally vulnerable, angered when she accuses him of merely exploiting her, and even a little touch erotic. This sexual element, however exciting, is not altogether justified. The Professor offers Eliza nothing more than a sexless ménage à trois, chums together with James Laurenson’s Colonel Pickering. In this delectable comedy of male-female relations sex remains off the menu.

Until 2 August. Tickets: 0870 060 6628, www.oldvictheatre.com/