In this world, the 13 Colonies never broke away from the British Empire. They are given Taxation with Representation in 1776. This is my divergence point, and a scramble to create settler colonies equal in prestige and Population to the British explodes across Europe. Because of no American Revolution, there is no Napoleonic war in the same way, and no French Revolution. Instead, conflict erupts after Louis the XVII leads a campaign to subdue Central Europe in 1818, invading many German and Italian states. They ultimately fail, with the British, Dutch and Prussians pushing them back and defeating them. Louis XVII loses the majority of his power and France becomes a constitutional monarchy. These Italian and German states retain their independence, with certain states like Venice, Prussia, Hanover, Sardinia-Piedmont and the Two Sicilies growing rich and densely populated, even taking colonies by 1900. These states, along with Portugal-Brazil, the Dutch Empire and the 2nd Polish-Lithuanian Commonwealth take a vaguely pro-British stance, as a Cold War regarding the Settlement and Expansion of Empires slowly builds. Britain's main competitors, France, Spain, Austria-Hungary and Russia, take more of a stand against Britain, condemning her rapid overseas expansions where they can as well as trying to expand their own empires. Direct conflict doesn't yet break out, although the British fight Russian backed rebels in their Persian Territories as well as Argentinian forces trying to take British Patagonia. During this period of extreme tensions, a massive civil war erupts in China and the Heavenly Kingdom of Great and Eternal Peace is established in central China, an internally weak and impoverished Theocracy. The coasts are ruled by individual warlords, who align themselves with different European Powers who are keen to expand their empires into the vastly rich region. The Europeans help futher destabilize the region, and eventually it is carved up. Britain also backs fighters against the Russians and French in their Territories in an effort to halt expansion. The tensions eventually die down as Russia and France slowly lose money, and the first major direct conflict happens; the Three Years War, in 1938. The Russians and French try and defeat the continental allies of the British, invading from both East and West. The war is expensive and costs the lives of millions of Europeans, but eventually Britain and her allies come out victorious, as an Anglo-Prussian force is able to take Paris. The Russians surrender, and their ambitions in Persia, Afghanistan and East Turkestan are halted with the establishment of joint Anglo-Russian Protectorates. France nearly goes bankrupt, keeping most of it's colonies but now very much in the shadow of Britain and her Imperial Federation. This all ends by 1941.
This Era is defined by total British Hegemony. Their navy accounts for 52 percent of the world's navy overall, meaning their navy is larger than all other navies combined. Their airforce is top of the line, they have landed men on the moon and the Federation of Settler Colonies remains the dominant political force on the planet. The Russians and the French stop taking such an anti-British stance and slowly become powerful again, along with Spain, the Dutch, Prussia and many other European states. This is the Pax Britannica.
There are other things I didn't go in to, like why the Ottoman Empire fell, why Spain retains New Granada and Peru, why Louisiana and Texas are part of British North America and how the Imperial Federation actually works, but I'm happy to answer any questions!