Two sisters, Thomasina Hanbury ("Thom") and Martha Hanbury ("Mars"), grow up in near total isolation inside a secluded English country house after becoming orphans. Thom is a brilliant inventor who creates a machine named Lola, named for their late mother, a device capable of tapping into radio and television signals from the future. The sisters mostly use Lola to listen to music from times to come, crafting a private window into a world they cannot access directly. Yet the calm of their existence is upended when the Second World War erupts.
As the Blitz shakes Britain, Thom and Mars begin listening to future news bulletins and broadcasting clandestine messages to warn civilians to take cover from impending German air raids, a daring act that saves many lives and draws the attention of the authorities. Lt. Sebastian Holloway of the Royal Corps of Signals seeks out the source of the transmissions and eventually brings his superior, Major Henry Cobcroft, to inspect Lola and its messages. Cobcroft is skeptical at first, but he agrees to let Holloway stay and help authenticate the signals. With Lola’s data, the Royal Air Force is able to shoot down a large number of German bombers, altering the balance of the air war and constraining Luftwaffe raids over Britain.
A romance blooms between Mars and Holloway, even as Thom grows increasingly enthralled by Lola’s power. She begins to banish Mars and Holloway from the house so she can work without interruption, convinced that her work with Lola can shape the future. The sisters start to witness concrete signs that their meddling with time is bending events, a realization that unsettles Mars and sharpens Thom’s resolve.
Mars’s curiosity about the future is tested when she tunes into a TV appearance by David Bowie from 1973, only to discover that the broadcast is hijacked by Reginald Watson Reginald Watson, whose music bears disturbing fascist themes. This moment strengthens Thom’s conviction to push Lola’s capabilities further, even as the consequences of their meddling begin to ripple outward. Thom orchestrates a risky trap for German U-boats in the Battle of the Atlantic, but the operation backfires in a grim way: the German plan succeeds in sinking the American ship SS Abraham Lincoln, killing around 2,000 people, and rumors spread that the British knew this attack would come, souring American support and delaying American entry into the war.
With the United States reluctant to join the fight, Germany makes a renewed bid to invade Britain. Thom receives signals that an invasion will land the next day at Southampton, only for the next day’s events to reveal the real landing at Dover—a deception that signals the first major crack in the sisters’ control over Lola. Cobcroft’s suspicions grow, and the sisters are arrested, tried, and sentenced to death as perceived double agents.
Just as the twins face execution, a German assault on London erupts, allowing Holloway and Mars to slip away to Cornwall. Thom is freed by German forces and becomes a public figure in a fascist government led by Oswald Mosley and Edward VIII, pursuing scientific progress as a tool for building a supposed perfect society. Adolf Hitler even visits Thom’s house during a wartime tour, but Mars detonates a bomb that injures Thom and leads to Mars’s capture and hanging in the front yard. Thom manages to rescue Mars but is fatally shot by a German soldier.
In the closing moment, Mars reflects that she made the film hoping it would reach Thom in the past and deter her from further Lola experiments. A final title card notes that Lola has never been found, and that the only known trace of the Hanbury sisters is a newspaper photo of them among a crowd celebrating V-E Day.
