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The Toclafane invasion would have occurred in the early 21st century on Earth, but due to the actions of the Tenth Doctor, Martha Jones and Jack Harkness, the invasion never took place.
Origins
The origins of the Toclafane Invasion lay in the last of humanity at the end of the universe trying to find a way to survive. They eventually tried to reach Utopia. With Martha, the Doctor and Jack's help, the scientists Yana and Chantho were able to activate the rocket to send the humans to Utopia.
The last of humanity boarded the ship and were sent away. They soon realised Uropia wasn't real; there we no "diamonds". They eventually crashed on an uninhabitable planet and regressed into the Toclafane. Sometime later the Master would return to offer them a chance to return to the universe and start again. (TV: Utopia / The Sound of Drums)
The election and invasion
The Saxon Master, on the day the Doctor an co. arrived, was elected Prime Minister of the United Kingdom under the name of Harold Saxon. He used this position to alert the humans to the arrival of the Toclafane and held meetings with a few of them to discuss their arrival.
The Doctor, Martha and Jack later found a way onto the Valiant and found out that at 8.02am the Toclafane were going to invade. They witnessed the murder President Winters by a Toclafane and Jack being temporarily killed by the Master's laser screwdriver. The Master Doctor aged the Doctor 100 years in human form and mocked him. The 6 billion Toclafane arrived and were ordered to decimate one tenth of the population. The result of their invasion crippled humanity, even Geneva was "slaughtered".
Martha teleported to Earth using Jack's vortex manipulator and witnessed millions of Toclafane wreak havoc. (TV: The Sound of Drums)
The Year That Never Was
Martha Jones spent the year in between May 2008 and May 2009 travelling the world. Alien races and invaders did not go to the planet because the human slaves were building missiles that would destroy them as ordered by the Toclafane. They patrolled the planet and would force workers to show identification or else be killed. The Toclafane also guarded the Master and the Paradox machine.
On board the Valiant, Francine, Tish, and Clive Jones were slaves of the Master. Jack Harkness was chained up in the engine room, and the Doctor remained on the ship living inside a tent and was forced to watch whatever the Master did. At one point he made them all stand on deck to watch as the Toclafane "burn[ed]" Japan.
Martha met with Tom Milligan, a doctor who was seeking to repel the Toclafane. They travelled to Professor Allison Docerty's lab and took a Toclafane hostage in order to discover the species' true identity. Upon capturing a Toclafane and opening up its armoured shell, they discovered that the Toclafane were the humans from the year 100,000,000,000,000 and the Toclafane that Martha captured had the memories of Creet, a young boy from planet Malcassairo whom Martha had met and liked. The Toclafane said that the legends had lied to them; that there were no diamond-skies in Utopia, only "darkness, the never-ending darkness" and then out of despair the humans regressed into sadistic, but child-like, states and integrated themselves into armoured shells, becoming the Toclafane.
They travelled with the Master to a void between dimensions, where they ultimately invaded Earth in 2008. When Tom Milligan asked why the Toclafane were slaughtering humanity, as humans and Toclafane were one and the same, the Toclafane laughed and said, "Cos it's fun." Tom, horrified, shot the Toclafane.
The next morning, the Master planned to launch the entire Toclafane fleet towards the Universe itself, as he had plans to rule the entire universe and create a new Time Lord Empire. Martha travelled to London to stop him, but she was caught by the Master, who had been aware of her presence. Thus she was taken to the Valiant once again and made to watch as the Master prepared to commence his war. He revelled in his apparent victory, until Martha revealed her plan that when humanity thought the word "Doctor", the power would make the Doctor whole again. This occurred and Jack led the guards to the TARDIS just as the Toclafane were about to leave Earth to invade the cosmos. Jack Harkness fought through the Toclafane guarding the TARDIS and destroyed the paradox machine. The Toclafane vanished from the skies of Earth and were teleported back to Utopia and thus they never invaded. Time reversed by one year and everyone forgot except for those aboard the Valiant. The only remaining impact of their invasion was the death of the president and the mental instability it inflicted on the Jomes family. (TV: Last of the Time Lords)