The only thing wrong with the British Museum’s rapturous trip through the Technicolor world of Utagawa Hiroshige’s prints is its final section, which explores this early 19th-century Japanese artist’s continuing global influence. A patchy sampling of Hiroshige’s imitators is all a bit rushed. But then, to do justice to his after-echoes would take a blockbuster in itself, not an epilogue.

Everywhere I looked up to this point, it was evident how precisely French impressionism followed Hiroshige’s cues. Take rain. It becomes a pleasurable urban event in Renoir’s The Umbrellas, but it was Hiroshige who first saw rain as a lighthearted excuse to put up umbrellas – in works such as his print Tarui, created in the 1830s. The impressionist theme of snow, enjoyed by Monet, is also delightfully anticipated by Hiroshige’s 1832-34 work Snow-viewing Along the Sumida River.

This triptych, in which a single scene is expanded across three separate prints in a favourite Hiroshige tactic, shows the French avant garde took much more than imagery from Hiroshige. Artists and writers in late 19th-century Paris adopted his whole philosophy. For Snow-viewing encapsulates the way Hiroshige looks at the world, with a hedonism he shares with the people he depicts. In this print, a well-dressed family hunch up in their robes on a cold day in Edo (now Tokyo). But they are not on a hard winter journey – they are just enjoying the way snow blankets nature.

Pleasure in the passing moment, from a shower of rain or fresh crisp snow to a restaurant meal or trip to the theatre, is Hiroshige’s ideal. His art wittily insists that happiness lies in savouring these little freedoms. This was the ideal the early European modernists took from his art, though what was calm common sense for this devout Buddhist would be for them a revolutionary escape.

This is, you might say, an exhibition about nothing. Hiroshige is a connoisseur of brief glances and weightless incidents. A man says goodbye to a female friend who is a sex worker, as the blue dawn shadows under a pinkening sky show that it’s daybreak in Edo’s sex district. In another triptych, two women watch their companion as she goes off to bathe. That’s it, but you could look for ever at their expressions, sinuous poses and colourful swathings of cloth.

Hiroshige portrays scenes so fresh you could restage them not just in fin de siecle France but London or Newcastle today. You thought pop-up restaurants were a 21st-century idea? Nah. In Enjoying the Evening Cool Along the Shijo Riverbed, crowds of people enjoy the temporary restaurants set up on a dried-up riverbed. In the foreground, men and women dining on a platform over the still-flowing part of the river laugh as one of friends does a comic dance. Here we go again: it’s Renoir’s Luncheon of the Boating Party. As for Manet’s Luncheon on the Grass, there it is in scenes of people picnicking on sheets spread out in the open air.

But Hiroshige did not live in modern times, nor is “modern” even a helpful term in relation to his art. He was born in 1797 in a Japan that had been ruled since the 1600s by the Tokugawa shogunate, a military dictatorship led by the samurai class that excluded virtually all foreign contacts. His 1830s triptych of a Samurai procession captures what his world was like, but with a twist: the ritualistic procession is almost exclusively female as a bride is taken to her elite marriage.

Maybe Chaucer is more relevant than Baudelaire. An 1851 print by Hiroshige depicts a fun-loving crowd of pilgrims heading for a mountain shrine by the sea, their bright costumes and dancing movements as robustly human as Chaucer’s Canterbury pilgrims. What makes this print, and almost every other here, so radiant is this artist’s ecstatic sense of colour. Seas like sapphire, skies on fire, acid reds and oranges, kimonos with myriad hues that contrast with the white faces of the women – it’s a kaleidoscopic trip. The matter-of-fact details of Hiroshige’s pleasure gardens, teahouses and picnics are irradiated by his cosmic colours. He sees nirvana in a blast of Prussian blue.

It was Van Gogh who was Hiroshige’s most passionate western fan. There are two versions here of The Plum Garden, which Van Gogh copied. In the different variants, Hiroshige casts the sky in varying tints of red-pink, as if the atmosphere is stained with plum juice. Van Gogh’s drawing for his canvas after Hiroshige is shown beside these juicy scenes. You feel his pained concentration as he carefully, spikily delineates the fruit trees, trying to drink the redemption and happiness of Hiroshige’s sweet strong art.