Honoré Lechasseur uses his newfound time-sensitivity to investigate a civil servant named Simon Brown, whose wife suspects that he is having an affair; however, his investigation is interrupted when Emily Blandish is found dead beneath Hammersmith Bridge, only to show up in person at the morgue while Honoré is identifying the body. Honoré and Emily are then assaulted by plainclothes policemen who warn them away from Simon Brown. Wondering if there’s a connection to Emily’s “death,” Honoré visits Simon’s mistress, Amanda, who speaks cryptically about the social implications of the atomic bomb. Honoré discusses the strange conversation with Emily, who agrees that the nature of warfare has changed now that a single bomb can wipe out an entire city.
Before they can come to any conclusions, they witness a gunfight in the street between a stranger and the plainclothes policemen who assaulted Honoré the previous night. Honoré intervenes and knocks the stranger unconscious, but when the police search him, they find that he has already passed on the papers they were trying to intercept. As Honoré wonders aloud where the man came from, Emily clutches his arm -- and they find themselves suddenly transported to a muddy field in the country in the middle of the night. As Honoré struggles to understand what’s happening, a tank rolls down the road, and he realises that he can see through it, as if it’s there and yet not there at the same time. When he looks at a nearby road sign, he sees the distance to Leeds measured in both miles and kilometres.
Honoré and Emily seek shelter in a nearby barn, where they find an old newspaper dated over 30 years in their future. As Honoré remains in the barn, struggling to come to terms with what’s happened, Emily visits the local town to investigate further. Her lipstick and clothing mark her out from the drab townspeople, and she is placed under arrest and taken to a nearby manor house for interrogation. There, she meets Radford, the man who was involved in the gunfight in 1949 -- but he doesn’t recognise her, and when he tries to “read” her past, he sees things he can’t explain. Emily realises that Radford is time-sensitive, and Radford is fascinated by Emily, who doesn’t seem to comprehend the way things work in his world, in which access to information is strictly controlled by a single political Party and people have forgotten the concept of historical change.
Honoré can still see two separate worlds at the same time, but he manages to focus on one of the worlds long enough to leave the barn and look for Emily. Outside, he runs into Amanda, who is on her way to the manor house. Amanda recognises him from 1949, and admits that she and Simon -- a translator in the Ministry of Defence -- tried to release state secrets about the atomic bomb to the entire world, in the idealistic belief that if everyone had the Bomb, no one would dare to use it. Instead, the people who held power acted to suppress the knowledge by imposing restrictions on education, burning universities and libraries and killing any scientists who could have built atomic weapons for enemy countries. Now there is only one Party, which controls how knowledge and information is disseminated to the people.
While discussing his Party’s policy with Emily, Radford gets carried away and sexually assaults her, excited by the first intelligent argument he’s had for years. Emily fights him off, and he pulls himself together; however, he then wishes while holding Emily’s shoulder that he could go back five minutes in time to stop this from happening, and suddenly he and Emily have done just that. Shaken, Radford sends Emily back to the manor house, where she realises what is to come, tricks her dull-witted guard, smashes her bowl of soup and hides a shard in her clothing.
Honoré reaches the manor house, where his time-sensitivity makes him aware that Radford is the one who will kill Emily and leave her body beneath Hammersmith Bridge. Honoré rushes to the manor house to rescue her, but is taken prisoner. When Radford touches him, both men “read” the other’s timeline, and Radford realises how Emily and Honoré came to his world. The time-sensitive is the navigator, and Emily is the pilot; together, they can travel through Time. Radford also sees himself killing Emily beneath Hammersmith Bridge back in 1949. Amanda then arrives at the manor house to give Radford a message which she claims Radford himself had given her in 1949. It’s a mission briefing from his own future self: Radford is to use Emily to go back in time to 1949, contact the younger Amanda, and pass on the secret documents that Simon will deliver to her. Radford, who can also see two contradictory timelines overlapping, realises that he must do so in order to ensure that his version of history triumphs over the other.
Radford forces Emily to “pilot” him back to 1949. They materialise beneath Hammersmith Bridge, but because Emily has seen her future body, she knows that Radford intends to kill her now. She thus attacks him with the broken shard of her soup bowl, and while he’s caught off guard, she speaks of Honoré. While the image is fresh in his mind, Emily grabs hold of them, transporting them back into Radford’s future. Emily releases Radford as they travel forwards through time, and he spirals off in an unknowable direction. Emily then materialises in Honoré’s cell in the manor house, and when they touch, they return to 1949, blowing apart the manor house with the force of their departure. This time around, Radford is no longer there to kill Emily or to pass on Simon Brown’s stolen state secrets, and the world that Emily and Honoré saw will not come to pass. However, they must now come to terms with their newfound ability to travel through Time together.