Barring the utter mauling of the Soviets by the Nazis and the liberation of Eastern Europe by the Wallies, here is the most plausible way to save Eurasian monarchies.
Europe:
Rumania: King Michael goes abroad and establishes a government in exile. After the fall of communism in 1989 he is invited to become a British style constitutional monarch.
Bulgaria: Same as above
Yugoslaiva: Have the Chetniks take a more anti-Axis stance and thus obtain greater access to Wallie weapons and support. Thus a civil war breaks out in post-World War II Yugoslavia, akin to Greece. Ultimately, the Chetniks prevail and a monarchy is reestablished or the country is paritioned between ethnic groups and between communist and anti-communist rebels. If Croatia becomes a monarchy then you've added to the total number of monarchies in existence.
Albania: Liberated by Britain (via Greece). The communists aren't imposed and the Brits return King Zog I to the throne. Alternatively, have the
attempted monarchist rebellions in 1946 prevail in toppling the communists or have the 1997 Monarchy referendum be won by the anti-republicans.
Greece: Avoid the 1967 coup in Greece.
Italy: Have the 1946 referendum go in favor of the monarchists.
Ireland: Remains a Dominion of Great Britain
Iceland: Remains part of Denmark or becomes a dominion (i.e. like Canada.)
MId-East:
Egypt: Avoid the 1952 Nasser Coup
Iraq: Avoid 1958 Coup.
Above two can be avoided by either improved economic situation, greater political liberalization (thus avoid the appeal of socialist factions), more pan-Arab inclination governments.
Iran: Avoid the 1979 Revolution, or the "
Black Friday." The latter will likely result in the establishment of a secular constitutional monarchy. The reform government would need to keep the Shah alive until his son was old enough to take over in 1981. Should he die before this, his wife would become regent and likely roll back the reforms.
Yemen: Avert the 1962 Coup. Either have greater British or Saudi support for Yemeni monarchy or have the Egyptians be unwilling to protect the republicans in the Yemeni civil war.
Afghanistan: Avert the 1973 palace coup and the 1978 Communist coup.
Asia:
Laos: Have the monarchist government prevail in the war against the communists
Cambodia: Avoid 1970 Coup by military and avoid the communists from turning on Sihanouk.
S. Vietnam: Either have the Japanese declared Kingdom of Vietnam remain in existence (i.e. no ROC and British occupation immediately after the war) or have Diem's rise to power (and the ensueing declaration of the abolition of the monarchy) avoided. Of course one would have to make sure that S. Vietnam remains intact during the Vietnam War.
Nepal: Avoid King Gyanedra's decision to impose direct absolute rule. This resulted in the anti-communist opposition and the Maoists cooperating to push for the abolition of the monarchy in 2008.
Tibet: Avoid have Tibet get incorporated into the PRC in 1950 (partially) or avoid the 1959 revolt that restulted in the total abolition of Tibetian government institutions by the PRC.
