^ That's correct - although there were certainly many monarchists in their ranks by default, the Whites were not universally monarchists. A lot of them were Social-Revolutionaries, Kadets and other democrats displaced by the Bolshevik coup we now know as the October Revolution, and several of the key White leaders (Lavr Kornilov, who worked closely with the Provisional Government until his chaotic maybe-coup in August of 1917 and seems to have favored an apolitical military dictatorship, comes to mind) were opposed to the Tsar. The most extreme absolute monarchists, the Black Hundreds, had lost a lot of credibility by this point.
The Russian succession is also muddled at this point: in general practically nobody wanted the specific original Tsar, Nicholas II, back since his reign had been rather disastrous to put it mildly, constantly battered by tragedy and failure literally from his coronation (a stampede at the celebration square killed over 1,000) onward. His only son Alexei was a hemophiliac and Nicholas already abdicated not only his imperial rights but also Alexei's to his brother Mikhail, who was killed by the Bolsheviks in Perm even before the main Romanov family. The remaining senior heirs, Grand Dukes Kyril and Nicholas, were at odds: although Kyril was more closely related to Nicholas II, he had pulled a Philippe Egalite move (submitting to the authority of the Provisional Government and even wearing a revolutionary red armband for a while in hopes that they'd make him regent or even emperor) which royally pissed off the other Romanovs.
Historically the Entente supported Alexander Kolchak to become 'Supreme Ruler' of Russia, and he was recognized as such by the other major White generals in 1919. As to the conditions of their aid for his regime, well to quickly quote his Wiki page:
Alexander Kolchak's Wikipedia page said:
On 26 May 1919, the Supreme Allied Council in Paris offered to provide Kolchak with unlimited supplies of food, weapons, munitions and other supplies (but not diplomatic recognition) provided that he was willing to meet the following conditions:
- Promise to convene the Constituent Assembly the Bolsheviks had disbanded in January 1918.[21]
- Allow local self-government in territories under his control.[21]
- Promise not to restore the aristocracy, the "former land system" and "make no attempt to reintroduce the regime which the revolution had destroyed" (i.e. not restore the monarchy).[21]
- Recognize independence of Finland and Poland.[21]
- Accept Allied mediation for relations with the Baltic states and in the Caucasus.[22]
- Promise to join the League of Nations.[22]
- Promise to pay all of Russia's debts.[22]
Pipes wrote that though the Allies wanted a Constituent Assembly to decide the future of Russia, they had decided in advance in their conditions that, for instance, there would be no restoration of the monarchy as well as many other matters that properly should have been decided by the Constituent Assembly.[22] Because Kolchak was entirely dependent upon supplies from Britain—the British had shipped him in the period October 1918-October 1919 about 600,000 rifles, 6,831 machine guns, and about 200,000 uniforms—he had to accept nearly all of the conditions.[22] In a telegram to Paris sent on 4 June 1919, Kolchak accepted every condition except for the independence of Finland, which he accepted only de facto, not de jure, saying he wanted the Constituent Assembly to grant Finland its independence.[22]
As the Allies were especially opposed to a return of the House of Romanov, Kolchak emphatically declared "that there cannot be a return to the regime which existed in Russia before February 1917."[22] The British War Secretary Winston Churchill pressed very strongly in the cabinet for British recognition of Kolchak's government, but the Prime Minister David Lloyd George would only do so if the United States likewise recognized Kolchak.[22] The American president Woodrow Wilson was strongly hostile towards Kolchak, openly doubted his word, and was against diplomatic recognition.[22] Wilson's main adviser on Russia was the former head of the Provisional Government, Alexander Kerensky, who told Wilson that Kolchak was a "reactionary" who would "inaugurate a regime hardly less sanguinary and repressive than that of the Bolsheviks."[23] Though American forces in Siberia co-operated with Kolchak, it was clear he was not the man favored by the United States as the next leader of Russia.[24] American forces had been sent to Siberia less to help the Whites than to prevent the Japanese, who had occupied the Russian Far East, from annexing it as Tokyo was openly considering.
Emphasis mine. (Also, obligatory fuck Woodrow Wilson for making yet another absolutely brainless foreign policy move and also fuck Alexander Kerensky, the archetypal sort of erratic and spineless liberal who opens the door of state to Communism) Long story short though, if the Entente is to have a major hand in suppressing the Reds, they won't allow the Romanovs to return. Most likely the 'Russian State' will limp along for years to come, with Kolchak - notoriously not a people-person and a terrible administrator - as a 'first among equals' while the other major White generals establish fiefdoms of their own (the viciously antisemitic Denikin, most pogrom-happy of the White commanders, will probably dominate in the south if Kornilov still dies, for example) and Entente aid helping them to avert a famine & slowly rebuild. Time will tell if any one of the generals can overcome the rest, become more or less Russia's Chiang Kai-shek (the indisputable leader of the nation, even if he has to put up with some remaining warlords) and lead the shattered country back to glory.
Big butterflies abound in the foreign arena with a White victory, as well. There being no Bolshevik victory in Russia takes a lot of the wind out of the Nazis' sails in Germany, for example. But before we deal with that, consider Russia's relations with the countries that became independent during WW1 and the collapse of the Empire. As you can see from the Entente's demands, they pushed hard for Kolchak to recognize the independence of these countries (chiefly Poland and Finland), which is definitely going to ruffle the feathers of the Russian nationalists among the Whites (again, such as Denikin, who historically attacked the Ukrainian nationalists of Symon Petlyura at the same time that both were battling the Reds). Other generals, such as the 'Black Baron' Wrangel (who reached an accommodation with the Georgians and Ukrainians) would be more receptive to such demands. Whether Russia makes crushing & reintegrating these countries after fully recovering from the RCW, the most obvious foreign policy maneuver it could undertake if it's aggressively-minded, will depend on which faction ends up on top.
