Tom Campbell, a London Special Constable, is on patrol near a jewellery shop. Men are burgling the shop and Tom is struck down by their getaway driver before he can stop them. Running to what appears to be a police box to call for backup, Tom enters TARDIS, a time machine operated by its pilot, Dr. Who, with his niece Louise and his granddaughter Susan. The Doctor pilots TARDIS forward in time to 2150, where they find that London is now an empty landscape of demolished buildings.
The Daleks, one-time adversaries of the Doctor, have invaded Earth and ravaged entire continents, while humanity's remnants have formed underground resistance movements. Some captured humans have been turned into brainwashed slaves called Robomen, but the majority have been taken to the Dalek mining complex in Bedfordshire, where the aliens' excavations extend to the core of the Earth. Louise and Susan are taken in by a group of rebels based in the London Underground, led by Wyler, David, and the wheelchair-bound Dortmun.
Meanwhile, Tom and the Doctor are captured by a squad of Robomen and taken on board a Dalek spaceship, where they are placed in a cell with a man called Craddock. The Doctor realises that the door is sealed by magnetism and breaks the connection with a plastic comb, but he is unaware that the escape is merely an "intelligence test" devised by the Daleks to determine who should be robotised. However, while the Doctor, Tom and Craddock are undergoing the conversion procedure, the rebels launch an attack with hand-held bombs.
During the battle, the Doctor flees with David while Tom and Louise, who is knocked unconscious by one of the bombs, stow away in a deserted part of the ship. The Daleks escape and take off for the Bedford mine with few prisoner losses. Wyler, having lost most of his team, returns to the rebel hideout, where Dortmun and Susan are waiting. The group commandeer a van to rendezvous with any remaining survivors in Watford, but Dortmun is killed by a Dalek patrol and Wyler and Susan are forced to abandon the vehicle before it is destroyed.
Deciding that the Doctor would avoid the Daleks in Watford, Wyler and Susan set off for the Bedford mine. David and the Doctor are also heading for the same destination, but are confronted by Brockley, an unscrupulous smuggler, who seizes their rifle in exchange for a promise to get them safely into the complex. The spaceship touches down at the mine. Tom and Louise exit the craft through a disposal chute and take refuge in a tool shed.
Meanwhile, Wyler and Susan shelter at a hut owned by a pair of spinsters who repair slave workers' clothes in return for freedom and food. However, the women betray them to the Daleks out of desperation. In the morning, David and the Doctor are brought into the mine by Brockley, where they are reunited with Tom and Louise. One of the miners, Conway, reveals that the Daleks are planning to drop a bomb into their mineshaft to punch out the Earth's core, which will be replaced with a giant motor enabling the aliens to pilot Earth to their Skaro home world.
However, the Doctor learns that the old shaft leads to a convergence point between the North and South Magnetic Poles. He deduces that, if the bomb were deflected down this path, the explosion's magnetic energy would be powerful enough to suck the metal Daleks into the core of the Earth. As Tom and Conway work to alter the bomb's trajectory, the Doctor orders David and Louise to create a diversion while he chooses to remain in the tool shed.
Brockley offers to help the Doctor and escorts him outside – where the scientist is not surprised to discover a detachment of Daleks waiting to take him away. The treacherous Brockley then tries to escape himself, but the Daleks destroy the tool shed with him inside. In the mineshaft, Tom and Conway run into Craddock who is now a Roboman. While fighting, Conway and Craddock fall to their deaths down the shaft.
Tom removes the timbers boarding up the entrance to the old shaft and then rushes back up to ground level. Sent to the Dalek command centre for extermination, the Doctor discovers Wyler and Susan. In the control room the inventor seizes an opportunity to distract the Daleks and commandeers the Robomen's command circuit, ordering them to turn against their masters. As the Robomen fight the Daleks, the Doctor escapes with Wyler and Susan, while the slave workers flee from the mine.
The Daleks defeat the Robomen's revolt and release their bomb into the shaft, unaware that Tom has successfully altered the route. The device is deflected into the disused shaft and detonates at the pole convergence. The Daleks, overwhelmed by the magnetism, are pulled into the Earth's core and destroyed. Meanwhile the spaceship, having just taken off, is brought crashing down onto the complex in a massive explosion. Later the Doctor, Tom, Louise, and Susan return to the past.
The Doctor materialises TARDIS a couple of minutes before the jewellery store raid, giving Tom time to take over the getaway car after knocking out the driver, then to knock out the other two thieves as they get into the back seat. As he drives off to the police station to deliver the criminals, he gleefully utters his name with the new title he hopes to be promoted to and laughs, "Detective Inspector Tom Campbell, OBE. The Doctor, Louise, and Susan wave goodbye as he passes.