I know that in the 15th century the system we'd recognize as a postal-service started cropping up, but back in the 4th crusade the Templars (and other orders like them) were still around and with them their internal messenger-system which seems to have been very effective. However I can't seem to find anything specific about either event only general statements on the speed of messengers (I suspect they might have hurried more with news this momentous however).
It wasn't spread by a postal service exactly, but by messengers and letters from Italian merchants and colonists in Constantinople and the Aegean Sea. Rumours spread after only a couple of weeks, but it took about a month for official reports to arrive.
The Genoese governor of Pera (the Genoese colony on the north side of the Golden Horn, opposite Constantinople) sent a letter, which arrived in Genoa as early as June 17. Meanwhile, the governor of Negroponte (Euboea, an island off the coast of Attica) heard the news on June 3. His letter to Venice on June 29, but they had already heard rumours there, presumably from the Genoese. The Venetians were eagerly awaiting this official report and they gathered outside the doge’s palace to hear the letter read, to grieve the loss, and to complain that the doge hadn’t done anything to stop it. There was actually a Venetian fleet already at sea on its way to Constantinople, but they turned back in June when they heard the news being sent back the other way to Genoa and Venice. The fleet arrived back home a few days after the letter from Negroponte.
The Venetians informed the Pope in a letter on June 30. Rumours had spread to Rome already as well, but the official report from Venice arrived by the end of July. I don’t think we have specific dates for when the news reached other places in Europe, but the pope announced plans take back Constantinople on September 30, 1453, so presumably everyone knew by then.
Greek refugees fled to Italy, France, and even as far as Scotland and Spain in 1454 and 1455. But everyone knew about the fall of Constantinople long before that. There were already numerous Greek refugees who had been living in Europe since earlier in the 15th century. Some had travelled there as refugees, or diplomats, or representatives at the Council of Florence in the 1440s, when the Greek and Latin churches were officially reunited (although this was extremely unpopular and impossible to enforce in Constantinople). So it’s possible that the news reached these Greek emigres first, but as far as we know, the official reports came from Genoa and Venice in June and July of 1453.
Sources:
Jonathan Harris, The End of Byzantium (Yale University Press, 2012)
Jonathan Harris, Greek Émigrés in the West, 1400–1520 (Camberley, 1995)
Michael Angold, The Fall of Constantinople to the Ottomans: Context and Consequences (Taylor & Francis, 2012)
Steven Runciman, The Fall of Constantinople, 1453 (Cambridge University Press, 1965, repr. Canto, 1990)