In "Nicholas & Alexandra," Robert Massey writes that one purported reaction among the peasants to news of the Tsar's possible death at Yekaterinburg was, "Merciful Christ, could they have burned the whole family alive?"
Since no formal announcement of the assassination was made, the world had only rumors to go on at first. But even rumors of the assassination of the Tsar and his family would naturally have evoked shock, horror, and revulsion among some, and indifference among others in 1918, just as they would today. However, the Imperial family were so removed from their subjects that little was known about them by anybody outside their immediate circle. Given that, and the turmoil Russia and the world were embroiled in at the time, I suspect that any reaction to rumors of the assassination would have passed quickly as people's thoughts turned back to the matter of daily survival.