It didn't take long. I'll use the example of Iraklie Tsereteli, a leading Menshevik who was exiled outside of Irkutsk in Siberia at the time:

Tsereteli was in a village called Usolye, which had a rail link to Irkutsk (70km away), which itself was the administrative centre of Siberia. The February Revolution started on February 23 (old style) with the protests against bread rationing in Petrograd. This news reached Irkutsk on the evening of March 2. Though it was initially suppressed by the governor-general, it got out, and by the evening word had reached Tsereteli in Usolye. He went down to Irkutsk and helped set up a local committee that became a de facto authority of the region, before ultimately leaving for Petrograd, arriving on the night of March 19/20 (reports differ).

Based on the above, it didn't take long for news of the Revolution to spread. While I note that the protests began on February 23, I should also be clear that it wasn't until March 2 that Nicholas II abdicated, and unfortunately what exact news was sent to Irkutsk is not made clear. However understand that there were telegraphs that went that way, which is how it largely spread across Russia (famously the message from the Mensheviks in Petrograd to Tbilisi was "Mr. Governmentson (მთავრობაძე, Mtavrobadze in Georgian) has died," a cryptic message to avoid the censors. So it didn't take long, though government authorities did try to suppress the news as best they could.

The reaction to it of course depended on where you stood politically. As noted Tsereteli immediately helped set up a Menshevik-led committee, which involved other exiles in the region. The situation in Tbilisi had the Mensheviks there inform the viceroy that he was no longer needed and they would take over. Cognizant of the situation, Grand Duke Nicholas, the Viceroy, peacefully left. Bolsheviks, many of whom were exiled in Russia or beyond, began to work their way to Petrograd. Lenin, exiled in Switzerland, was initially dismissive of the whole revolution, angry that the Bolsheviks had not led it, though by the end of March he was negotiating with the Germans to get himself back to Russia (via the famous "sealed train").