Exciting news! After many seconds of careful consideration and absolutely no discussion with the other administrators, I have decided that this week is European Empire Week!
The great European Empires lasted for centuries, and shepherded the world from being the sparse, simple, agrarian world of the medieval times to the sprawling disaster of the modern age! They have therefore had a considerabe effect on modern history and therefore even a tiny divergence to just one of these empires could lead to a big change to the world at large.
The most expansive European Empires included the British Empire, the French Empire, the Spanish Empire and the Portugese Empire. Here're some ideas;
British Empire Never Fell
This history is almost as classic as 'What if Hitler won the Second World War?'. The British Empire actually lasted right up into the Cold War when its power quickly waned after the onslaught of two World Wars, the Great Depression, and the economic chaos that ensued after the close victory by the Allies, leading straight into the Cold War. This gives plenty of opportunity for the British Empire to hold onto power (though some would say that its time had come either way).
There is also the possibility of a resurgence in the British Empire later in the twentieth century; the modern-day Commonwealth of Nations is a political organisation that covers most of the terratories owned by the British Empire. The novel Resurrection Day is an alternate history in which there is a resurgence of the British Empire in a world scarred by nuclear war.
None of the European Empires Fell
Why just the British Empire? What about a world where the Britain, France, Spain, Portugal, Belgium, the Netherlands and Italy still compete for power all over the world?
If you're thinking of making a more Embellished history for this one, expect lots of steampunk!
In the Medieval period the Catholic Church had an immense influence on Europe, owning approximately 10% of the continent's wealth at one point! What if the Vatican was even more influential, uniting its supporters such as Italy into a kind of theocratic Papal Empire, founding colonies along with the rest of the Europen Empires?
It is likely that modern times in such a history would be significantly less technologically advanced, due to the fact that the Catholic Church would often be unhappy with the introduction of new ideas (Galileo was just one example).
Spain and Normandy were the two great European powers of their day. Spain "discovered" and colonised much of the New World before these assents were taken away by other European Empires later on in history.
How would history have changed if the two united under a common interest?
Treaty of Tordesillas
The Treaty of Tordesillas was a treaty made by the Spanish and the Portugese in 1494, which basically said that the New World belongs to the Spanish (all of it), and any new lands discovered in the Old World belong to the Portugese. The fools! Though I don't exactly see how such a thing could take place, what if the Spanish and Portugese managed to maintain these claims on basically the whole world outside of Europe?
I'm sure there are many other (much better) ideas that people come up with. Notice how I haven't mentioned how the above histories could come about, I'll leave that to you.
I hope this has at least given you a spark of inspiration for future projects!