One of the highest-grossing movies of 2023, it also won awards for leading actor Cillian Murphy, who portrays the American theoretical physicist J. Robert Oppenheimer, supporting actor Robert Downey Jr, editing, cinematography and original score.
Nolan, who won his first Bafta for directing, thanked his cast and crew in his acceptance speech.
"In the real world there are all kinds of individuals and organisations who have fought long and hard to reduce the number of nuclear weapons in the world... in accepting this I do want to acknowledge their efforts," he added.
Like Nolan, Murphy had been favourite to win his category and in his acceptance speech, he referred to the man known as "the father of the atomic bomb".
"Oppenheimer was this colossally naughty, complex character and he meant different things to different people," Murphy said.
"One man's monster is another's man hero. That's why I love movies because we have a space to celebrate and interrogate and investigate that complexity."
Emma Stone picked up the leading actress award for sex-charged Gothic comedy Poor Things, which won five prizes overall.
Da'Vine Joy Randolph won the supporting actress prize for her role in The Holdovers, a comedy set in a boys' boarding school.
The Zone of Interest, about the commandant of Auschwitz and his family living next to the Nazi death camp, won three prizes - outstanding British film, film not in the English language and sound.
Courtroom drama Anatomy of a Fall won the first prize of the night, original screenplay. Adapted screenplay went to comedy-drama American Fiction, which is based on 2001 novel Easure by Percival Everett.
Best documentary went to 20 Days in Mariupol, journalist Mstyslav Chernov's personal account of the siege of the Ukrainian city in 2022.
"This is not about us, this about Ukraine, about people of Mariupol... the day before yesterday another Ukrainian city has fallen... many cities before that," Chernov said in his acceptance speech.
"So the story of Mariupol is a symbol of everything that happened, a symbol of struggle."
As well as the celebrities attending the ceremony at the Royal Festival Hall by the River Thames in central London, the guest list also included Bafta president Prince William.
Known as the Baftas (British Academy of Film and Television Arts), the ceremony was hosted by actor David Tennant.
