AS others have noted, the best possibility would be during the Russian civil war, with Alaska becoming a White refuge of sorts. On the other hand, since the WW1 allies invaded Russia in 1918/19 to assist the whites, I could easily see the USA, Britain, or both jointly occupying Alaska with little intention of restoring it as a fully independent state, even if it becomes a League Mandate. A lot would depend on how Alaska developed as Russian territory between the 1860's and 1900's. If it became home to a sizeable and largely self-sustaining White Russian population, immediate independence might occur after the Whites lost. If the Russians did not settle the territory in large numbers in the 19th century and it remained largelly native (Inuit, Aleut, and American Indian), I suspect it would either be assimilated into Canada or divided between the US and Britain.