Couldn't you say that OTL in 1898 met this criteria (Britain, France, China, Russia, USA)?, But since this is an AH board I offer this (ca. 1900), not in order:
1. British Empire (including OTL British Empire as well as what OTL is the USA to the Mississippi River and Canada minus BC.
2. Russian Empire (including OTL Russian Empire as well as Alaska and western coast of North America to Columbia River)
3. Mexican Empire (OTL New Spain to the Isthmus of Panama)
4. Chinese Empire (OTL Chinese Empire)
5. Islamic Caliphate (OTL Ottoman Empire at greatest extent plus North Africa and Islamic areas on western/central africa
This would leave Japan, some of Oceania, and small areas of subsaharan Africa independent, none of which would equal the smallesr
If this is in 1900, I should point out that Mexico only had 13 million people, or about half the population of Spain (at the time). Even if you include all of the rest of new Spain, that's still about 60 million, tops (about 2.8% of the global population in 1900). Germany had more people than that. The Ottoman Empire was also relatively small, with only about 30 million people (less populous than Italy).
Population in 1900:
1. British Empire: 419 million
2. Russian Empire: 132 million
3. "Mexican Empire": 60 million (OTL New Spain)
4. Chinese Empire: 415 million
5. Ottoman Empire+Egypt: 42 million
Combined, that gives you about 1068 million out of 1700 million people alive at the time, or about 63% of the global population.
Still, this might be possible with an ancient PoD. The American Revolution never happens, establishing British control over North America. France and Spain are still crushed in some kind of early 19th century war, establishing British naval dominance. Belgium never gains independence, but instead ends up part of France. Germany ends up uniting into some kind of barely-stable Grossdeutschland. South America ends up getting re-colonized. Japan never manages to modernize, and is partitioned among the great powers. The Ottoman Empire goes to war with one of the Great Powers and is defeated and partitioned*.
Korea is annexed by China. Siam is annexed by France. Persia is partitioned between Britain and Russia. Russia annexes Bulgaria. Britain annexes Nepal. Mexico and the Central American states become British protectorates. We're only at 88%, and I'm out of ideas. So let's have France, Britain, and Germany partition Italy, because nobody's done that in a while. France gets its OTL Napoleonic Italian territory (minus Venice), Germany gets northeastern Italy, and Britain takes southern Italy.
So, in 1900, you've got:
1. British Empire: 611 million
2. Chinese Empire: 465 million
3. Russian Empire: 187 million
4. German Empire: 150 million
5. French Empire: 120 million
This gives us a combined population of 1,533 million people out of a global population of 1,700 million, or 90.17%. The 6th most populous country is Sweden, with 5 million people (less than 5% of the 5th largest country, France). Of course, this will only last for a few years before all the overextended empires collapse.
*For both the Japanese and Ottoman partitions, I'm assuming that the 4 European powers all get equal shares of the population.