What was the last possible moment the following monarchies could have been saved?
Russia
Germany (all of em)
Austria-Hungary
Let's see.
Putin appears to be interested in the old Romanov Family as a means of claiming royal heritage. Having a house of Romanov-Putin might happen, its hard to know.
The Hohenzollerns are still around today, but I doubt they'd be able to retake the throne of Germany after WW1--and there was some nastiness in WW2 where one of them said: Where Hitler leads, a Hohenzollern can follow.
The German Laender have various royal families dating before the reunification. If they take views hostile to Nazi Germany, they might be supported by the Allies if they decide to regain some power. They'll never be anything more than figureheads in a post-ww2 Germany.
Austria and the Hapsburgs is one of the real fascinating post-war stories. After the war, MI-6 was in detailed communication with Otto II, whom probably would have been returned as Kaiser if France and the United States agreed and the Political situation in Austria demanded it. (Karl Renner didn't need this)
As for political continuity:
The Russian Czar simply had too much power, and Russia was too autocratic to survive as effectively an absolute monarchy surrounded by nations more democratic to itself. Nicholas II refused to reform the country--and was forced to abdicate as a result. His brother was Czar for a day but did nothing with his powers. Unless the Whites win the war with a Romanov behind them, the Monarchy in Russia is gone.
Germany's Overthrow of the Kaiser stems from Wilhelm II's lack of military aptitude and his reliance on his top officers to keep the army behind them--which they didn't. Assuming that these rebellions did not occur, signing the Treaty of Versallies would be the kiss of death for his reign.
Austria-Hungary is interesting because this is the one the Hapsburgs could have continued on. There were several coup attempts against Admiral Horthy in favor of Otto Hapsburg, and if things had played differently at least one part might have kept the Hapsburgs going.