As We Rise: Photography from the Black Atlantic review – every image contains some kind of magic

How did a Canadian dentist become one of the most important collectors of photography in North America? The story begins with one picture. Specifically, a James Van Der Zee picture of a young couple dressed in raccoon fur coats and posing with their Cadillac V-16 Roadster in Harlem. Everything is gleaming. It's the ultimate aspirational image.

But this picture was taken 1932, the era of racial segregation in America. Slavery had been abolished 67 years previously but anti-black racism, including lynching, was rife, and rural southerners flocked to northern cities to escape it. Harlem, New York, was in the throes of its renaissance , as Black culture from jazz to literature flourished. The unnamed couple in the photograph, solemn and proud, embody that movement's gumption, what Tina M Campt has called the ability of Black artists to "twin pain, trauma, loss, with Black people's virtuosity in survival, and the capacity to inhabit tremendous joy in spite of all these things."

Ten-year-old Kenneth Montague, a first-generation Jamaican Canadian, was astonished when he saw the photograph at the Detroit Institute of the Arts in the 1970s. Decades later, he bought it. It was his first acquisition in what is now the Wedge Collection – Canada's largest privately owned collection of Black artists, and one of the most prominent private collections of photography worldwide, numbering some 400 works.

The Wedge Collection was once literally wedged into the nooks and crannies of Montague's home, hence its name, but at London's Saatchi gallery it revels in the space it gets. As We Rise: Photography from the Black Atlantic is a capacious and glorious display, loosely organised into four themes but intently focused on intimate depictions of Black subjects.

It could be read like an extended family photo album – one that draws on a sense of shared situations, sentiments and solidarity between Black people that breaks free of geography and time. But what a family album. The foundations are the African modern masters of studio and street portraiture. A large, silver gelatin print of Malick Sidibé's 1963 Nuit de Noel (Happy Club) allows you to get closer to the details – not just the unstoppable energy of the elegant dancing couple but the young woman who sits in the background, hands folded in her lap, overlooked, or waiting her turn on the dancefloor.

Oumar Ly's rural studio portraits are lessons in improvisation and ingenuity – the hands and heads of assistants holding up textile backdrops mustered from whatever happened to be around. The urgency is at odds with his subjects' refined composure, held for a moment in the photographer's gaze.

Where the major names appear, it is with atypical or unexpected contributions. A single rare print by Carrie Mae Weems is a small but gripping self-portrait taken in 1975 when the artist was 21. She turns her back to her camera in a dramatic pose, showing a torn T-shirt, inviting us to travel to the details of the room she stands in. It's a precursor to her later, groundbreaking works in which small gestures convey everything. Weems' picture also anchors the importance of intimacy in these pictures – personal kingdoms, mausoleums to lives lived on the mantelpiece.

The show is perfectly paced, moving between places and generations, legendary photographers and some who are barely known. After four vast rooms, portrait photography feels like a superpower of self-projection. There is not one image here that doesn't contain some kind of magic.

Surveys of the Black Atlantic are usually dominated by voices from the US and the UK, but As We Rise includes important photographic legacies from Canada, the Caribbean and Brazil. A pair of pictures by Afonso Pimenta are a glimpse into an archive documenting life in Aglomerado da Serra, one of Brazil's largest favelas, in the 1980s. Pimenta, who lived in Aglomerado da Serra himself, took more than 300,000 pictures there in that decade (80,000 or so survived, but they were stashed in storage at his home until 2015).

One of Pimento's pictures is of a young mother, Claudia, with her two young children, her smile scintillating as much as her sequined dress, standing in front of a cabinet proudly displaying neatly arranged possessions. Across the room, it forms a fascinating dialogue with Deana Lawson's festive fiction The Coulsons, in which a mother with two children smiles next to an artificial Christmas tree. Whether staged or real, and where or when it was taken, slowly becomes irrelevant in these pictures. The desire is the same: to see, or conjure into being, an ordinariness that is radical.