Pete's World
Pete's World is the Doctor's name for an alternate universe the TARDIS accidentally fell into. (DW: Doomsday)
Pete's World is similar to the normal universe, though slightly more advanced. At the time of the Doctor's first arrival, much of the world's technology is provided by Cybus Industries, a company run by John Lumic. This includes the Zeppelins and EarPods. Politically, Britain became a republic sometime after 1879, which led to the election of a British President, the instigation of a curfew and the banning of alcohol. (DW: Rise of the Cybermen)
History
Much of this worlds history is unknown though it is suspected to mirror our own in nearly all ways bar Britain being a republic, a curfew and banning of alcohol, air travel being done by airships instead of planes and the introduction of a company called Cybus which apparently sped up that worlds technology to be around 10 or so years ahead of our own. In addition, the Doctor later learns that time on Pete's World moves somewhat faster than in the "normal" universe; as a result, although mere months pass between his first (DW: The Age of Steel) and second (DW: Doomsday) encounter with inhabitants of the parallel Earth, a couple of years had passed on Pete's World.
The Doctor was accompanied by Rose Tyler and Mickey Smith during his first visit to Pete's World. Rose learned that, on this world her father, Pete Tyler, had not died in an automobile accident (DW: Father's Day) but was still alive and a hugely successful businessman and commercial pitchman connected with Cybus Industries. On this earth, most of the populace is "wired in" to the Cybus network via cellphone-like earpods that upload information (including a daily joke) into the user's brain. Mickey found that his parallel version, Ricky Smith, is a member of an anti-Cybus terrorist cell called the Preachers. This version of Mickey is more rough and violent than he is, and Mickey later learns that Ricky is also gay (DW: The Age of Steel - deleted scene included on the DVD).
Rose's mother, Jackie Tyler, was a socially active diva in this world, and both lived in a mansion. Rose learned to her chagrin that she did not exist on Pete's World, but the Tylers did name their dog Rose. (DW: Rise of the Cybermen).
It was at this time that Lumic unveiled his version of the Cybermen and tested them out in hopes of "upgrading" the human race. In the ensuing conflict, Great Britain's president is killed, as is Jackie Tyler when she is upgraded and becomes a Cyberman, and Ricky Smith.
After the defeat of Lumic, Mickey chose to remain on Pete's World to take over (most) of Ricky's life, and primarily to care for his elderly parallel-grandmother. (DW: The Age of Steel)
Months later, a Torchwood 1 experiment involing a rift in London results in an unusual form of contact with what is believed at first to be "the other side" (i.e. Heaven); in fact, a rip in time and space was being opened which allowed the Pete's World Cybermen (now autonomous after the death of Lumic) to cross over and begin an invasion of Earth, with the intent to "upgrade" the human race. Unknown to Torchwood, a few Cybermen came across first and slowly began infiltrating the facility, converting Torchwood personnel.
As the invasion began, the Doctor and Rose encountered fighters from Pete's World who had travelled over to try and stop the Cybermen invasion: Rose was reunited with Mickey, while the Doctor was reacquainted with Pete Tyler and the Preachers. The Doctor made a brief visit to Pete's World during this encounter, long enough to learn that the rips in time and space caused by the movement of people and Cybermen between worlds, exacerbated by the Torchwood rift experiment, were causing catastrophic and potentially civilization-ending damage to the parallel world. It was around this time that the Doctor dubbed the parallel earth "Pete's World" after Pete Tyler. (DW: Doomsday)
During the ensuing battle (which was made more complicated by an attempt by the Cult of Skaro to revive the Dalek Empire and simultaneously battle the Cybermen), Roes, Jackie Tyler (from our universe), and Mickey were ultimately trapped on Pete's World, with no way safely back without destroying their planet. Some months later (as reckoned on Pete's World; an unknown time later in our universe) The Doctor was able to make one final tearful communication with Rose by utilizing a supernova. Afterwards, no further communication was possible. The Doctor learned that, among other things, Pete's World had its own version of Torchwood. (DW: Doomsday)
Unknown to the Doctor, Rose and others on Pete's World began working on a "dimension cannon" to allow transport between the two earths. During this time, it was discovered that, for reasons unknown, the stars in the sky were going out (a phenomenon termed "The Darkness"). Learning that this effect was likely affecting parallel worlds and other dimensions, and after possibly trying unsuccessfully to contact the Doctor via a video signal (DW: The Poison Sky and Midnight, Rose used the new technology to transport herself back to her original Earth on several occasions, on one occasion arriving too early (DW: Partners in Crime), and on another occasion arriving in an alternate timeline in which the Doctor had been killed (DW: Turn Left). She finally reunited with the Doctor in the "main" universe (DW: The Stolen Earth). She was soon followed by Mickey Smith and Jackie Tyler.
After defeating the coming of "The Darkness", Mickey chose to remain on his original Earth, and the Doctor returned Jackie and Rose to Pete's World where he asked Rose to stay in order to look after the new Doctor. (DW: Journey's End)
It remains to be seen whether the doors between our universe and Pete's World are again closed forever, or if the technology used to transport Rose and others will be used again to allow access to and from Pete's World.
Errors
If the Torchwood in our world was set up by Queen Victoria in response to The Doctor then that means there must have either been a Doctor in that universe, which seems unlikely as such a person has never been mentioned, or something else happened to the Queen to make her develop the institute. Even if the Doctor was not present for those events in this universe the tragedy and paranormal activity could have inspired the creation of Torchwood regardless. The Torchwood of this universe could have also been created under completely different circumstances.
In light of the events in Journey's End, it is possible that this Torchwood was formed in response to the "New Doctor" sent back with Rose. This, however, would require that the event leading to the creation of Torchwood is a point very much fixed in time, and presumably also that the New Doctor creates or finds a method of traveling in space and time. Their Torchwood seems to have a pretty good start on this, based on their ability to send Donna back in Turn Left, so it is absolutely a possibility.
Alternate versions of inhabitants of this universe
Pete Tyler was a successful business-man in this universe, though he had never had any children. He ran Vitex, a subsidiary of Cybus Industries.
Ricky Smith worked with the Preachers and was a wanted criminal. The Preachers were trying to stop Cybus Industries' experiments on the poor, which were actually early attempts to make Cybermen. (DW: Rise of the Cybermen, The Age of Steel) Unfortunately, Ricky died, leaving the Preachers without a leader. Mickey, Ricky's double from the other universe, joined the Preachers, and Jake became the new leader.
