Germany: Pretty easy: win WWI; other scenarions include: German old elites restores the monarchy in the 1930s to counter Hitler and Germany losses WWI, but before the German Revolution, at least only Wilhelm II abdicates.
Qing: After 1900? I'd say impossible, by then the Hundred Day' Reform was rejected, China lost against Japan, losing their place as the big dog of the East and the people's faith on the Manchu was pretty low.
Russia: Bit hard, winning WWI of course helps, but Russia still had a lot of problems related to authoritarism and Nicholas was an idiot, the Empire is in desperate need for reforms and the leadership wasn't reallt keen on it. Maybe after winning the people start pressuring for more freedom thanks to their sacrifice in the Great War. Another funny possibility is losing WWI earlier, this way Nicholas abdicated, but the monarchy isn't necessarily doomed.
Italy: Monarchists win the referetum, it was a close call.
Portugal: Don't kill the king, simple.
Greece: The king stays out of politics, denouncing the coup would improve his image.
Bulgaria, Romania, Hungary, Serbia: Monarchies are restored after the fall of communism (unsure about Hungary).
Austria: Win WWI. Should the empire eventually dissolves in ethnical strife, the Habsburgs remain as Kings/Archduke of Austria inside the German empire.
Poland, Finland, Baltic States, Ukraine: CPs wins WWI and put German monarchs on those countries.
Ireland: Better handling on the Irish Question (not sure how).
Turkey: Either win WWI or stay out of it.
Arab monarchies: No Israel might save them.
France: Oh boy! I can see only one wild, remote, possibility: Central Powers win WWI, get land from France and all that jazz, this leads to a revival in Monarchism in France, leading the a restauration (probably from a main-line Orleans or a Bonaparte).
Vietnam: The US backs someone with more IQ to be their puppet in South Vietnam. The north probably still wins though.
Laos: The French are clear in who is going to be king.
Afghanistan: No 1970s coup.
Iran: No knowledge on the Islamic Revolution... The shah less of a dick?