The only Countries of the World worth really mentioning are the Monarchies, because they are the only countries that have any sort of magical power to them. A lot of these countries in the BBP world are different from what you're used to. Since Magic was codified along with Science, the power of a Monarchy exists in the Monarch marrying the Land, thus becoming one with it. By and large, democracies are not near as stable and lasting as monarchies in this world, though the Founding Fathers of the U.S.A. found a way around that.
Note: Due to the current instability of the middle east, I'm going to have to do a lot of research to determine how different things are in the web comic. Keep in mind, Socialism never got off the ground in the BBP comic, though Islamic Extremists are still a bit of a problem.
Afghanistan - The Durrani Empire gained power under Ahmed Shah Durrani . Dost Mohammed Kahn failed to gain control of Kabul, and was put to death by Zaman Shah Durrani, the grandson of Ahmed, and the 3rd king of Afghanistan. The Khan tribe never gained control of Afghanistan, and is ruled by the Durrani to this day. Due to the fact that Russia never went Soviet thanks to the quick thinking of Grigori Rasputin, Afghanistan was never invaded by a modern world power, and Afghanistan retains their autonomy. Today the king of Afghanistan is Satsa Shah Durrani.
Albania - Gjergj Kastrioti Skanderbeg , the legendary hero of Albania, also known as the Dragon of Albania, freed Albania from Ottoman rule in 1443. He established the new Crown of Albania, the Dragon Crown of Illyria, named for their ancestors. The Dragon Crown of Illyria still exists today, and is still held by a Skanderbeg. Illyrian paganism is still practiced to this day, and the Dragon King is seen as the spiritual keeper of the Law of the Gods (though the Dragon King does not speak for the Gods themselves, nor do any priest). The Illyrian paganism, unlike the greek and roman that it's compared to, is monistic, meaning the Illyrian philosophy is one where one sees divinity and sacredness in all things, not just those socially considered "good" or "holy".
Andorra - Same as in our world.
Belgium - Same as in our world.
Brunei - Same as in our world.
Cambodia - Same as in our world.
China - Since Russia's Bolshevik Revolution was unsuccessful, and Karl Marx was a playwright, the concepts of Communism and Socialism were never really explored. The Mao Dynasty took over the Qing Dynasty. The current Emperor of China (after some very short reigns following Mao Zedong 's death) is Hu Jintao . The Mao Dynasty is different from previous dynasties, as the successors are not chosen from one's family, but from the Mao Party council. It is unclear if the Mao Dynasty will ever return to bloodline successors or not. The Guangxu Emperor, the Last Emporer, had not left any heirs, thus there is no threat to the Mao rule. The Mao Dynasty began embracing Capitalism after the succuss of Hong Kong and its reclaiming purchase from the British. However, the Mao Party Council is little more these days than figureheads for the Han.
Croatia - The first King of Croatia, Tomislav of the Trpimirovic dynasty, was crowned in 925. King Petar Kresimir the IV brought the kingdom to his medieval peak. His son, Petar Kresimir V refused union with Hungary in 1102. To this day the Kingdom of Croatia is ruled by a monarch. The current Monarch is Stjepan "Stipe" Mesic , whose mother was found to be of royal blood when Stipe was young.
Denmark - Same as in our world.
Japan - Mostly the same as in our world, the Japanese Emperor actually has a great deal of political, social, spiritual, and magical power.
France - France decided to adopt a Constitutional Monarchy via popular vote in 1988. However, since the three claimants (Legitimist Louis Alphonse, who sites Salic law for why the Orleanist claimant is not legitimite, Orleanist Henri Count of Paris, who sites that descendants of King Philip V of Spain became illigitimate when he took the Spanish throne, and Imperialist Charles Napoleon, of obvious imperial lineage) would not relinquish their claims to the throne. The French Constitutional Council upheld both that Salic law of succession must be upheld, and that descendants of King Philip V of Spain were ineligible for succession. However, contrary to the Legitimists' claims, the Constitutional Council ruled that legitimate heirs to any former French monarch (aside from King Philip V of Spain who had abdicated the French throne) could pretend for the throne. Because of this, the French Constitutional Council ruled that Charles Napoleon was the only legitimate successor to any previous French throne. Emperor Charles Napoleon was coronated in July of 1990, and was married to the land in accordance with international magical law and French tradition. The Emperor's position is purely ceremonial, as the power still resides in the Constitutional government. (Please forgive the lack of proper accent marks. If I get a chance to edit this on a Mac, I'll put them in.)
Lesotho - Same as in our world.
Liechtenstein - Same as in our world.
Netherlands - Same as in our world.
Norway - Same as in our world.
Romania - Present day Romania encompasses 16th century Wallachia, Transylvania, and Moldavia. The King of Romania, Dmitri Tepes, is a direct descendent of Vladimir Dracula (rom for "The Son of the Dragon") Tepes (The Impaler). While Vlad Dracula Tepes was born in Transylvania, he ruled Wallachia. Despite the attempts of the Vladislav I, II, and III to destroy Vlad Dracul (Dragon) and Vlad Dracula Tepes (Son of the Dragon, Impaler), it turned out that Vlad Dracula Tepes had become a vampire (from his wife), and learned the secret to intentionally making vampires. Thus he forced his children to go through the change, in order to make them stronger leaders as well as immortal. To this day, the Tepes Family's elder brothers take turns being the King of Romania, so that the others can go out and have fun. Lately, Dmitri's been stuck with kingship for a while. (The Tepes Family (Dmitri, Antonio, etc.) are owned and copyright Laughing Gryphon Productions characters.)
Russia - After the defeat of the Bolshevik Revolution, Nicholas II of Russia found evidence (provided by now ICoM member Grigori Rasputin) that proved that rogue nobles were conspiring against him to enrage the citizens and incite the Bolshevik Revolution. Nicholas II was too concerned with his family and not concerned enough with his country to be a good Tzar. However, he was also kept so very out-of-the-loop by his corrupt advisors, that the atrocities of his reign passed without his knowledge. When Grigori brought the requisite proof of these things, he ordered a public execution for the corrupted nobles who were willing to destroy Russia just to get it out of Nicholas II's hands. Grigori performed the magical execution himself, and appointed his young and now fully healed son Alexis (1920), at the ripe young age of 16, to investigate the affairs of Russia, to see who needed help, what works needed to be done, and how to prioritize the work of the empire. In the same year, Anastasia Romanov became her father's foreign affairs advisor. Both Alexis and Anastasia were groomed into these positions by their mother Alexandra, Grigori Rasputin, and Nicholas II himself. Alexis took over as Tzar of Russia in 1933. He reigned until 1998, when his son began his reign as Peter IV, the current reigning Tzar of Russia.
Saudi Arabia - King Abdullah bin Abdul-Aziz rules unthreatened.
Siam - Prajadhipok, Emperor of Siam, gave an impassioned speech to his people about the finances of the country, and instead of events turning out like they did in this world, the people rallied behind him. Because he was married to the land, he knew that a coup was in the works, so he pretended to go on vacation, and set a trap for the betrayers. He handed them over to the people, and the people were not merciful. Prajadhipok, also known as King Rama VII, went on to participate in alliance with Great Britain and France in WWI, and eventually brought about the democratic changes he had been working for. Siam became the first democratically responsible Empire, ruled by a reigning Emperor. In 1943, he passed succession to his son, named Mongkut after the fourth of the Chakri dynasty, now King Rama VIII. The King of Siam has brought his people prosperity, security, and knowledge to the modern day. King Rama IX reigned from 1943 to 1989, and King Rama X is the current King of Siam.
Spain - (Requires Research)
Swaziland - Same as in our world.
Sweden - Same as in our world.
Thailand - See: Siam
Tonga - Same as in our world.
United Kingdom and its Commonwealth - Same as in our world.
United States of America - The biggest difference is that the Founding Fathers found a way to put the power of a King into the office of the Presidency. The Oath of Office represents a magical geas to the Gods, to not only defend and uphold the Constitution, but to take the land as one would a spouse and lover, to hold and protect with all one's might. The U.S. Constitution states that taking the Oath of Office is a sacred act that renders the President as being married to the land for the duration of his term(s). Not only that, but the Consistution is a magical contract in this universe. Any President who tries to fundamentally change the nature of the United States, to pervert its freedoms, will go insane after leaving office, or will die in office of illness or karma. Yes, Karma is often fatal.