Peckham, boasts the South London Gallery which is located there, is sometimes called “Little Lagos” because of its strong connections to Nigeria’s biggest city. In this exhibition Lagos transforms Peckham enticingly: its sounds, sights and even flavours infuse the gallery.

Emeka Ogboh has worked particularly hard to layer the two cities together: his Lagos Soundscapes fill this London space with African urban street sounds, not just honking traffic but chants, street cries and rapping bus conductors. This recording was even played to Ogboh’s own craft beer – displayed in the show as a sculpture in crates and on sale at the cafe – as it fermented in south London to “flavour” it with the sounds of Lagos, as well as Nigerian spices. It is named after a tough Lagos saying: “No Food for Lazy Man”. And the magic realist brewing method works, for it tastes outstanding.

This is fun, but Victor Ehikhamenor takes you deeper. This great contemporary artist delves into Africa’s religious and artistic heritage astutely, and his new work for this show is typically startling. From the “outside”, facing into the gallery, you see a kind of chapel, a white facade with a red gothic door, covered in pale, pearl-like Catholic rosary beads and little crosses. It is a closed, even chilly edifice. But go behind: the curved yellow space you find hidden here is decorated not with Christian symbols but cowrie shells, and hosts a low wooden altar lined with pairs of carved wooden ibeji – twin – statuettes. An unusually high number of twins are born to Yoruba people and twins are regarded as sacred and powerful. If one twin dies, carvings such as these of the deceased child are treated like real babies.

In Ehikhamenor’s compelling installation the emblems of Christianity on the “outside” give way to a hidden world of Yoruba religion and art on the “inside”. It’s a suggestive image of African history and being. It might also be a nod to the Lagos-Peckham connection, suggesting this London district is far away yet indelibly connected.

That twinship is triumphantly asserted by Adeyemi Michael’s film Entitled. You can sit in cosy armchairs by a table loaded with snacks, evoking his childhood home, to watch this film of his mother dressed in Yoruba regalia, including a gold dress and vast violet head-wrap, riding a horse through the streets of Peckham. She gracefully maintains her posture and controls the horse’s pace (her steed also starred in Peaky Blinders) as people look on in baffled awe.

Living with pride may be a British Nigerian style. It’s certainly the comic achievement of Yinka Shonibare’s 1990s art classic Diary of a Victorian Dandy, from which two scenes are shown here in their huge gilded frames. It’s always a delight to survey how elaborately Shonibare pastiches William Hogarth’s visual narratives such as The Rake’s Progress and Harlot’s Progress. His expansive photographic tableaux set in stately homes cleverly capture the deep perspectives of Hogarth’s scenes and their use of symbolic objects, yet with a modern, unresolved ambiguity. The rakish Black dandy stands heroically as white Britons adore him.

Where Ehikhamenor takes you on a journey into Nigeria’s cultural heritage, Shonibare ironically takes on the history of British art. Recently, however, he has been strongly engaged with Nigeria and last year opened his non-profit organisation Guest Artists Space, providing residencies for artists both in Lagos and on a working farm near Ijebu Ode, Ogun State. A room in the gallery’s Fire Station annexe showcases collages and drawings done by London artist Chiizii on her research residency there, mixing Igbo history and pop culture.

This show’s tale of two cities cuts between continents in a memorable collage. Studio portraits of Lagos, denizens from an already nostalgic past, glow in a lightbox while Christopher Obuh’s No City for Poor Man explores in empty, depopulated photographs the sci-fi spaces of Eko Atlantic City, the “Dubai of Africa”, that is being built in Lagos State. Will anyone ever live there?

Meanwhile Temitayo Shonibare sits on a London Overground train wearing an orange wig that completely covers her face. Nobody says anything and most avoid staring. It’s an image of both the freedom and anonymity of London life. The surrealist wig was made by Peckham hairdresser Solomon Paramour.