Code Geass takes place in a timeline where things went really well for British imperialism; the Celts kicked Julius Caesar and the Romans off the island, Elizabeth I had male heirs, "Washington's Rebellion" failed... all with the cumulative effect of making "Britannia" the only superpower on the planet as of the early 21st century. They were defeated by Napoleon, though, forcing them to abandon Britain and relocate to what had been the colonies — that is, North America, which is entirely under their control. Napoleon's Europe-spanning empire also laid the foundation for a faux-EU before he died (likely of poisoning by one of Queen Bessie's spies). It bears mentioning that there are certain factors present in the world of Code Geass that indicated that perhaps it Never Was This Universe. Such as Sakuradite and Geass itself.
Full Metal Panic! follows a present day where the Cold War never ended (due to Mikhail Gorbachev having been assassinated, and thus, the political reforms which led to the breakup of USSR never came to pass) and the arms race led to combat mechs on the battlefield — aided along by Black Technology, which comes the Whispered, rare individuals gifted with latent Psychic Powers.
By the end of the series, it's revealed that the villains' plan is to undo the alternate history, changing key events to turn reality into a peaceful one more in line with our own. The source of the Black Technology was, in fact, echoes caused by the Whisperer using her powers to probe backwards in time to find which events needed to be changed.
The finale of the manga version of Chrono Crusade reveals that it falls under this genre. Demon's homeworld—a spaceship/fish/...thing called Pandaemonium—is called out of the depths of the Atlantic ocean by Aion, which causes a tidal wave that destroys New York City. Chrono goes after Aion to try to stop him, and they end up fighting in Pandaemonium. At some point in the process, it blows up, creating a ring around the Earth that's visible in the sky even in the 1990s.
The anime ending averts this by trying to stick to OTL. It's implied that Aion shot John Paul II.
The Kerberos saga takes place in a world where Nazi Germany won in Stalingrad. It eventually leads to a total Axis victory... in Europe. Japan still falls in 1945. But it's the Germans who occupy and morph the country into a fascist dictatorship.
Read or Die takes place in a British-dominant world, complete with hidden superpowers. By the sequel, however, the Empire had completely collapsed, leaving the world at the mercy of various secret organizations and the United States.
In at least some early Universal Century timelines, it's implied that the circumstances leading up to the founding of the Earth Federation itself was due to an alternate outcome of the Cold War. This could also apply to Mobile Suit Gundam Wing, given the show's rather Anachronism Stew atmosphere and the After Colony calendar starting in the 1970s.
Strike Witches takes place in a world where aliens invaded in 1939 before World War 2 and forced the world to unite in order to stop them. Also the only way to stop them are by using magical imbued schoolgirls wielding rocket legs with characters like Winston Churchill and George S Patton making cameos. Officially the alternate history goes back further, with BC standing for Before Caeser, the German monarchy never losing power, and giving Japan a more active role in past events.
Jin involves a Japanese neurosurgeon from 2000 being transported back to 1862. He introduces germ theory (nevermind that the germ theory was actually proposed by Girolamo Fracastoro as early as 1546 and with the works of Louis Pasteur and Robert Koch alreadywas pretty much the mainstream by 1862) and modern surgical techniques to Japan and the West, among other medical ideas. Butterflies abound from his actions. Among the results? The Meiji Restoration doesn't take place and the Tokugawa Shogunate survives.
.hack diverges initially around 2002 with the founding of the UN's World Network Commission in the wake of mounting cybercrime. But the real changes happen when a virus called Pluto's Kiss is unleashed on December 24, 2005, crippling thousands of computer systems worldwide and effectively crashing the Internet. This provided the ALTIMIT Corporation virtual monopoly over rebuilding the net and helped set the stage for The World.
Concrete Revolutio: Choujin Gensou gives superpowers and giant mecha to everyone, and plonks them squarely after World War II. Political commentary and superpowered battles ensue.
In Tokyo Ghoul, the titular species has existed alongside humanity since at least the 19th century, although no one is sure where exactly they came from. The CCG, an organization created to investigate Ghouls, was founded in 1890.
Gintama takes place in a world where sufficiently advanced aliens made contact with Japan before the Black Ships did, causing the country to modernize much faster than they did in real life.
In Astra Lost in Space, we're led to believe that its world history diverged from ours sometime in the 1960's, with the Cuban Missile Crisis resulting in WW 3 and wiping out a big chunk of the world's population, leading to massive changes in the structure of the world. Though later revelations seem to suggest that there's something up with that history, given that Lina's time frame would set Earth's destruction a mere six years before the plot kicks off.
A Certain Magical Index is more of a case of Never Was This Universe due to the existence of magic, but special mention goes to the United States being on its third Hispanic President, the G14 having been established, a second Costa Rican civil war having taken place, the existence of the Elizalina Alliance of Independent Nations (a union of nations and countries formerly part of Russia) and a non-nuclear power United Kingdom having retained its long-held animosity against France (in part because they were forced to give up their nuclear arsenal to France by the European Union).
As of NT 9 it is explained that the current reality the series takes place in is the result of reality being destroyed and recreated millions of billions of times, each with its own slight alterations to history and the varying laws of physics that make it different from the previous one. So it is very possible that the universe is still ours but the fabric of reality within it is not any longer.
Combined to an extent with Unintentional Period Piece (as it was made and released during The '90s but takes place in 2015), the In-Universe backstory of Neon Genesis Evangelion was that in the year 2000, a global cataclysm called the Second Impact occurred, killing off large swaths of humanity in addition to causing all sorts of damage to civilization. The most visible Unintentional Period Piece aspect is the tech used in the show (no smartphones, Shinji listens to music on a tape player), which can be justified by how said global cataclysm slowed down development in certain ways (and probably killed off the inventors of said tech).