Actually, I'd argue that Bungou Stray Dogs is set in a timeline that very much parallels our own!
You're not wrong that Bungou Stray Dogs is its own timeline, but there are actually WAY more indicators of period inspiration than only World War I and closer connections between Bugnou Stray Dogs' timeline with ours. Rather than a completely separate world, Bungou Stray Dogs seems to be set in an alternative timeline in which the existence of skill users dramatically changed the global geopolitical landscape. And I know that because it's actually possible to roughly, and with several exceptions, trace the current Bungou Stray Dogs timeline to the 1930s. (Hence the Guild being dressed like 1930s Americans!)
The older characters in Bungou Stray Dogs are based on Meiji-era authors (namely Natsume, Fukuzawa, Mori, and Hirotsu). The Meiji era of Japanese history extended from October 23, 1868, to July 30, 1912, and was characterized by Japan shifting from isolationist feudalism to modernization and industrialization based on Western scientific, technological, philosophical, political, legal, and aesthetic ideas. (Otherwise, Japan was at risk of colonization by Western powers; it was a transition undertaken under duress.) That's why Fukuzawa is aesthetically a rōnin (the actual Fukuzawa was born in 1833, decades prior to the Meiji era), why Mori rattles off Western foreign political strategies with which Fukuzawa is unfamiliar (the actual Mori was born just six years prior to the Meiji-era), and, most importantly, why Yokohama is sometimes referred to as the Yokohama Settlement, and why it is implied that there are national powers in Yokohama. (For example, in 55 Minutes, an antagonist declares, "If the Yokohama Settlement and the various nations' military parties within it are wiped off the face of the earth, there will be no way for their governments to hide the truth.")
The Yokohama Settlement was established in 1858, through treaties with the US, England, France, Russia, and the Netherlands that opened several cities and five ports to those Western powers. Although the treaties were abolished in 1899, the influence of those treaties appears to persist in Bungou Stray Dogs' current timeline, implying that they did exist, but perhaps not with those specific powers, and perhaps not within the same time frame. (This is further supported when Chuuya debuts in the story's current timeline; he mentions to Dazai that he had only just gotten back from quashing trouble in the West, which may be referencing the western section of the Yokohama Settlement.)
The younger characters (such as Chuuya, Dazai, Atsushi, Kunikida, Ango, etc.) were all part of literary and poetry scenes characterized by the impacts of World War I and World War II on Japan. Although their ages relative to each other and the older characters are NOT the same as their real-life namesakes, they roughly correlate with the generational differences cultivated by Japan's transitions from feudalism to industrialization to post-World War I.
Further, World War I was called the Great War, which is also what Bungou Stray Dogs calls the ability-based war that transpired 15 years prior to the current canon timeline. Like the real-life authors, the Bungou Stray Dogs characters are consistently and constantly grappling with the impacts of the Great War, which created orphans (like Atsushi and Akutagawa), traumatized soldiers (like Mimic and Fukuchi), advanced and unprecedentedly destructive methods of warfare (in WWI, it was chemical warfare, in BSD, it's ability-users), and which prompted multinational treaties and organizations that sought to prevent such devastation from happening again (Standard Island in 55 Minutes; in the manga and anime, it's implied the United Nations exists, although the logo is just slightly different from the real-life United Nations, which was not formed until after World War II).
As you've pointed out, there are still SEVERAL differences between the real-world and Bungou Stray Dogs' timelines. Many of these can be explained by the existence of skill users, such as the mixed technology in the series, which ranges from the 1980s to the current day. Others are analogs, such as Verlaine and Chuuya, who were both created in response to the Great War, such as how WWI rewrote how wars and war technology were approached. Others are less easily explained by anything other than the butterfly effect, such as the seeming continuation of Japan's port treaties and, as mentioned above, how instead of the League of Nations, the aftermath of the Great War in Bungou Stray Dogs led to the creation of the United Nations, decades before the United Nations formed in our timeline. Further, in Bungou Stray Dogs, France, Germany, and Japan were allied in the Great War, with Japan withdrawing early due to Fukuzawa's intervention (France and Japan were allied against Germany in WWI). The most inexplicable difference is that one of the Western foreign policy strategists cited by Mori is Henry Kissinger, who was only a child in the 1930s. So on, and so forth.
However, that it parallels our timeline explains exactly what you've noted above, which is the reason why Europe is the center of geopolitical power. In the late 1800s and early 1900s, that was still very true, especially in the aftermath of World War I (and which would continue to be true if Europe had powerful ability users). It also explains why the most powerful ability user group, the Order of the Clock Tower, serves as the Queen of England's personal guard-- while the monarchy isn't Britain's head of government now, it was in the early 1900s. It also adds context to Season 5, Episode 61 of the anime, in which Fukuchi reveals that his actions were spurred by his learning that there would be another global war, one much more destructive and violent than the prior. In other words, Fukuchi sought to prevent the Bungou Stray Dogs' version of World War II.
And so while you're right that Bungou Stray Dogs exists in its own timeline, it's not fundamentally different from ours, and in fact, follows the same patterns and themes that the real-life character namesakes explored in their bodies of work. Following the logic of Bungou Stray Dogs existing in an alternate but parallel timeline, it roughly takes place in the early- to mid-1930s. However, it's a version of the 1930s in which skill users exist and have existed for long enough for their skills to shape the course of human history. Thus, the timeline in Bungou Stray Dogs is a sandbox of alternative history that nevertheless honors the ideas, traumas, and eras of the authors whose lives and literary works inspired the series.
That being said, it's a fictional universe subject to the author's purposes, so nothing is certain unless indicated by Kafka Asagiri. But understanding the historical parallels certainly explains a lot, and provides a nifty guide for understanding the underlying literary references.
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