League of Extraordinary Hogwarts Students takes place in 1864 in a Victorian England that is not quite as we know it.
For one, we have all the assumptions of a hidden "wizarding world" that we find in the Harry Potter novels; a world where people with magical talent have been taking care of their own for many years. Queen Victoria and the Minister of Magic, Tobias Smoot, have a cordial relationship, but your average muggle Londoner knows nothing about this world. On the other hand, wizards and witches tend to know more about the "muggle world" than their counterparts in Harry's day might.
This game, in fact, takes place at Hogwarts in 1864, where Phineas Nigellus, ancestor of the Most Noble House of Black, sits as its Headmaster. As in Harry's day, tensions between pureblooded wizarding families and muggle-born wizards and witches are present; the tension is heightened by the Headmaster's own favoritism of purebloods.
One might call this world steampunk, in that steam has been used to further the technology of the day beyond what where it was in our 1864. In this world, for example, several steam-powered prototypes Charles Babbage's Analytical Engine have been built. Each is as big as a room, but some lucky universities and scientific societies have one. One even found its way to Hogwarts. The railway project in Britain has been fast-tracked by Queen Victoria, and steam-powered locomotives are already motoring to Cornwall, Manchester, Aberdeen, and other locations. Steam-powered flight is a reality, even if a relatively new one. The first transatlantic steam-powered flight will be tested in just a few more months.
Medicine is another area where one might find great advances over the 1864 of history. The barbaric process of bloodletting has been almost completely eradicated, for example. Louis Pasteur, who in our world wasn't concerned with germ theory until the 1860s, has already done his best work, and sterilization procedures have improved all over Europe. Many lives have been saved as a result of just these two advances.
With the Antikythera Mechanism being put to its original use--meteorology and astronomy--again, the accuracy of weather forecasting has also improved. Most troubling these days is a summer season that, across the North Hemisphere, still hasn't faded. Well into this season, sweltering heat is the rule of the days, and meteorologists don't foresee a change anytime in the future.
In this crazy weather, and with the increasing dependence on technology, some might say that God, or the gods, have passed from this world. Perhaps this is why so many cults have sprung up recently, all claiming to have the secret to eternal life. This is something that has not escaped the attention of wizards and witches, who suspect the work of some malign sorcery and are working to right the situation.
Even these small changes from the 1864 we know have created a "butterfly effect" on the fictional tapestry of this world, such that fictional characters and dates associated with them might not line up with those in our world. Patience is advised in navigating this familiar, and yet very foreign, setting.