Burberry and Celia Birtwell are both national treasures of sorts. One is Britain’s grandest luxury name, the other the textile designer who dressed bohemian 1960s London in her prints and has a place in art history as David Hockney’s favourite muse.

Burberry’s designer, Daniel Lee, has always loved Birtwell’s prints. When the two met this spring, he fell for “her energy, and her zest for life”, and a one-off Burberry meets Birtwell collection that closed London fashion week was born.

Burberry’s calling card is a graphic check; Birtwell is all blowsy doodled florals and psychedelic snaking vines. “The check is so dominant and so brilliant that at first I thought: what can I do here?” Birtwell said before the show. She hand-painted a less regimented version, which Lee loved. “She made the check more human. If you are buying a luxury product it’s not a necessity, it’s an emotional purchase, so it needs to feel human.” A doodle of round, Smartie-bright faces from her archive, drawn 20 years ago but never produced, became another print of the season.

The mood of the show, held around the corner from Burberry HQ at Chelsea School of Art, was buoyant: charming prints, soft silhouettes, and the eclectic guest list that is as much a Burberry signature as the check itself: actor Fiona Shaw and footballer Djed Spence on the front row, with rapper Central Cee making a cameo on the catwalk.

Burberry is bouncing back from its crisis years, steered by its chief executive, Joshua Schulman, and designer Lee. This summer, the brand reported its first revenue growth in three years, as retail sales rose 5% year-on-year in the second quarter. Schulman’s strategy is offering “value for money in a luxury context”, which translates as pivoting from an ill-fated attempt to sell Burberry handbags at Hermès prices to more realistic price tags. Lee has brought a youthful, gen Z inflection to the Britishness that is Burberry’s calling card.

A successful element of its turnaround has been Burberry zeroing in on the outerwear market, where Burberry has bona fides to best every other name. British heritage is a trump card when it comes to coats, and the appeal of the Burberry name reaches from arctic explorers to football fans. For this summer collection, trenches came in slinky silk taffeta, with puffball hems and ruched collars; one model carried an embroidered umbrella, against summer showers.

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To amplify its London fashion week message, Burberry is staging a small display in an upstairs gallery of the V&A museum in South Kensington, which will be open to the public until 3 January. The Burberry Trench: Crafting an Icon includes a military overcoat worn by a lieutenant colonel in the British army in 1939, designer Hardy Amies’ own Burberry trench, as well as metallic leather and gold snakeskin runway trenches from recent catwalk shows.