They'd probably try to model themselves on the British system, with Grand Duke Nicholas chosen as the new Tsar. He would largely serve as a figurehead and a rubberstamp to a resurrected Duma.
However, the Duma would probably be fairly corrupt and divisive, and I can see the military and nobility trying to continue their dominance of playing a big role in the government, a la Weimar Germany, with the ever-present threat of a coup if things begin to get out of hand in their eyes.
Denikin, Wrangel, Kolchak, Krasnov, and maybe Yudenich would probably dominate the early political scene, along with some people like Milyukov, Guchkov, Ryabushinsky, and some of the less-militant Mensheviks.
However, this regime would most likely be terribly unstable and corrupt, again like Weimar Germany. If we are talking post-WWI Soviet borders, you are going to see irredentism crop up big time, forming the core of some sort of fascist movement based on the principles of "Autocracy, Orthodoxy, and Nationalism" like the Mladorossi movement. Similar to the Nazis, they would most likely be extremely xenophobic, outright reactionary, and extol some sort of corporatist platform (Mladorossi's slogan was "Tsar and the Soviets").
Chances are that, once Nicholas III dies in 1929, Cyril becomes Tsar. He was a noted supporter of Mladorossi and other White Emigre groups that had fascistic bents.
So you'll probably see a collapse of parliamentarianism by the early '30s and the imposition of a fascist-like government, probably headed by Kazembek, who was very popular and charismatic, though it would be more akin to Mussolini's Italy than Hitler's Germany.
There are of course huge butterflies given a Red defeat. We may very well not see Hitler rise without a communist bugbear to stir up crowds (anti-Semitism was a big part of the Nazi's platform, but anti-Communism was a bigger vote/support getter).
Anyways, you'll probably see a really antagonistic relationship with Poland and the Baltics. If Hitler or some other right-winger does come to power in Germany, I can certainly see the two powers making plans to divide Poland along pre-WWI lines.
As for the Jews, I can certainly see them being left largely unmolested by the earlier administrations, but a long history of Russo-Ukrainian anti-Semitism isn't just going to disappear, especially since most of the populace blamed the Jews for near-everything, especially communism. They'd probably be blamed for the Great Depression, too. They will probably suffer the same fate as their Weimar cousins; not extermination, but government-sponsored pogroms and discrimination, as in the Tsar's time.