OTL, a week after the Imperial family was murdered (17 July 1918) the town of Ekaterinburg fell to the Whites (25 July 1918). Is it possible for the Czechslovak Legion to get there early. I don't say that they would ncessarily rescue the emperor etc (as the Bolsheviks falsely believed), but I could imagine it eould be a massive PR boon to the Whites to have the Imperial family and a blow to the Bolsheviks to have lost them.
Let's assume that the imperial family don't get killed in the evacuation as was Alexander Beloborodov/Nikolai Gorbunov's official plan.
What happens next?