The morning after the 1852 election, the New York Times declared Franklin Pierce the projected winner, using election results as turned in to telegraph services such as the Associated Press.

Since that election, the New York Times has projected the winner for every presidential race in the next morning's paper, except in races where it's been too close to call. In those years, the Times has reported the latest election returns as of the time of printing, and readers could follow the action daily until a winner was declared.

Before that, you wouldn't have to wait as long as you'd think. Up through the 1844 election, election day was different in every state, and were held over the course of about a month. So election results would trickle in, and you could follow along in the newspapers to see where the latest vote totals had come in from and who was in the lead.

For instance, the 1836 election was held between November 3 and December 7 of that year, and if you look at issues of the New York Post between those dates (behind a paywall, sorry), there's a section of the paper that's dedicated to the latest election tallies.

They called the election for Martin Van Buren on December 9, two days after the last voting took place. Though it was more than a month after the first votes had been cast.

Obviously, if you lived outside a major city, it would take longer before you heard of the election results.

EDIT: Here's the actual article announcing Pierce's win, at least an extract from it. The full page linked above on the NYTimes website is from a day later.