I know "White Russia" doesn't mean fascist. That's why I said "a fascist White Russia."
If they do not succumb to fascism, (though I think that, given the era and the condition Russia would find itself in, it is entirely plausible, even likely) I think you'll see either a weak central government with stronger regional ones (like what was developing under Yeltsin), or a strong centralized government along an Ataturk line, like Kerensky was sort of going for.
And I would posit that the Polish would be no less hostile to a White Russia than a Soviet one, especially if the Whites tried to mount something like the Polish-Soviet War (which could perhaps occur if the Whites win early enough; I'm not sure the French or British would say "boo" if a White Russia retook Poland since I believe it was Wilson that called for an independent Poland, though I could be wrong). And if Pilsudski comes to power, he would in all likelihood not have any sort of diminished Russophobia; he pretty much believed that Russia would always represent a threat to Polish sovereignty.