Torchwood existed on Pete's World.
As with the Torchwood Institute of N-Space, they were based at the Torchwood Tower in Canary Wharf, where they discovered the breach in the Void. In contrast to their counterparts, this Torchwood's research advanced to the point that they developed devices that allowed transport to N-Space. Ultimately, the People's Republic of Great Britain discovered their research and took control of their facility. However, the Cybermen infiltrated Torchwood, and used the facility to transport their five million strong army to invade the Earth of N-Space. The process of transporting so many Cybermen took them three years, at which point they arrived in N-Space in 2007, while the Preachers, led by Pete Tyler followed them from Pete's World in 2010.
Through a dimensional transporter, the Tenth Doctor was introduced to the parallel Torchwood by Jake Simmonds, where he warned the Preachers of the damage the technology caused to the wall between worlds. As a result of the battle which saw the Cybermen defeated as the Doctor trapped their army in the Void, Rose and Jackie Tyler of N-Space were trapped in the Torchwood of Pete's World as the breach between worlds was closed. Months later, however, the Doctor was able to make temporary contact with Rose, who announced her intention to join Torchwood to defend the world that was now her home. (TV: Doomsday)
Torchwood developed the dimension cannon which was able to use weakening barriers between universes to punch holes in reality, enable travel to parallel universes again. Rose insisted on being the first volunteer to use it. (AUDIO: The Endless Night)
Behind the Scenes
Accoding to reference books, this version of Torchwood was created by the president of Great Britan after Queen Victoria was visibly biten by the werewolf as the Doctor was not present. (REF: Aliens and Enemies)