Remember that the Constitution leaves it up to each state to decide how to choose its electors for the Electoral College in Presidential elections. And in those earliest elections, there was no uniformity of a popular vote. Instead, there were three ways that electors were chosen:

• By the state legislature (the legislature could appoint electors all favorable to a single candidate, or they could appoint electors in some proportion favoring opposing candidates)

• By a statewide popular vote (usually winner-take-all, like most states today)

• By a popular vote decided by Congressional district (so some districts' votes could go to electors favoring one candidate, and other districts' votes could go to electors favoring a different candidate)

In states with some kind of public vote, the polls would typically stay open for many days if not weeks, for qualified voters to make the trek to their polling place to cast their vote. The actual date that the electors of the Electoral College met and cast their votes was (and is) designated by Congress, so all the public votes had to be cast and delivered to the state capital by that date. Likewise, in states where the state legislature elected the electors directly, they had to appoint them by the date the Electoral College was scheduled to cast their votes.

Once the Electoral College votes were cast in the state capital, then the results had to be transported to Washington, D.C.

One other thing to be aware of: the Election Day(s) for Congress and Governors in each state might not be the same day(s) as for the President.

Because of all this, there was actually quite a lot of information that could be reported by the press even before the Presidential polls closed. For example, the Congressional elections might already be over in a state, allowing newspapers to predict how the Presidential election was going to result when the polls closed for that state. Or in states where the legislatures picked the Electoral College electors, legislators might tally up which of their colleagues were going to vote for who, and what the likely result was going to be, well before the official vote was cast. They might then leak this info to the press. And even in states where there was a statewide popular vote, the polls might close several weeks before the Electoral College met, allowing preliminary totals to be reported in the press.

As a result, the election results were reported in the press more rapidly than might be expected, given the transportation limitations of the time. I'll give two examples: the 1800 election, and the 1808 election.

First off, the 1800 election is a bit of an outlier because it resulted in a tie vote between Thomas Jefferson and Aaron Burr. Even so, it only took about three weeks after the Electoral College cast their votes for the results to be first reported in the press.

One source tells me that the earliest polls in the elections opened back in April, though another source says that they didn't open until October. Either way, the National Intelligencer newspaper (the Washington, D.C., paper that was something of the Washington Post of its era) could report some of the Congressional election returns in Pennsylvania, Maryland, Delaware, and New Jersey by October 31, 1800. Based on those totals, the Intelligencer predicted that Jefferson was going to carry most of the votes from MD, PA, and NJ, while Adams was going to carry DE (the only one of four being a winner-take-all state that year).

On November 3, the newspaper made a more elaborate prediction based on their knowledge of all the states, predicting an Electoral College win of 81-41 in favor of Jefferson over Adams.

Still, the Electoral College that year wasn't set to cast their votes until December 3. And again, unlike today, since so many of the states didn't rely on a winner-take-all popular vote to determine the electors, anything before the Electoral College met was only a very loose prediction.

So it wasn't until after December 3rd that anything official could be known. The results were hand-delivered to Congress thereafter. The earliest results arrived from Virginia and New Jersey exactly one week later, on December 10. One week after that, on December 17, the National Intelligencer could report the official results of all but three states, the stragglers being Vermont, Kentucky, and Tennessee. No doubt, these were the stragglers because they were landlocked states not adjacent to Washington, D.C., without easy access to the Atlantic Ocean to deliver the votes by ship.

By then, it had become apparent the election was much closer than the National Intelligencer had predicted in early November. As it stood on December 17, Jefferson was ahead 66-51, with 70 votes needed to win. And Jefferson was predicted to win both Kentucky and Tennessee, which would put him over the top.

And that's what happened. By December 24, Tennessee's and Vermont's votes had arrived in Washington, D.C., with Jefferson up 69-55. At worst, Jefferson was guaranteed a tie, though Kentucky had always been considered a "safe state" (as we might say today) for Jefferson.

Kentucky's votes finally arrived in the Senate on December 28, delivered to Jefferson himself. As Vice-President at the time, he was also the President of the Senate. So, the results were known exactly 25 days after the votes were officially cast, on December 3.

Of course, Jefferson and Burr tied, but that was another issue. On February 11, the Constitutionally-mandated joint session of Congress met to officially count the Electoral College votes from the delivered envelopes, which made the tie betwee, Jefferson and Burr official. With that done, the Constitutionally-mandated "contingent election" was conducted in Congress, with Jefferson emerging as the winner on February 17 after many ballots over several days.

News of his election likely took about the same time to reach the various state capitals as it took to deliver the Electoral College envelopes to D.C. — between one and four weeks. Of course, some people found out sooner (like those living in Virginia and Maryland in proximity to Washington, D.C.), while others probably found out later (like towns in relative isolation on the frontier). Even so, with news this important, it would have spread quickly, and anybody interested in knowing likely found out within about six weeks of the Congressional vote on February 17. A considerable number of Americans likely found out by the time Jefferson was being inaugurated on March 4.

The 1808 election wasn't so messy. On November 7, 1808, the National Intelligencer ran a convenient table listing the Election Day in each state, and by what mode the electors in those states were being chosen. The earliest Election Days had already passed by then—November 4th. The last was South Carolina's legislature meeting to choose electors on December 6. The Electoral College was mandated to meet in each state capital on December 7.

In that article, the Intelligencer predicted a landslide victory of James Madison over Charles Pickney. They were right. Just twelve days after the Electoral College had met, on December 19th, the National Intelligencer declared James Madison the President-Elect. Thirteen of the seventeen states had delivered their results to Washington, D.C. by that date (the stragglers being Kentucky, Tennessee, Ohio, and Georgia), giving Madison a total of 101 Electoral College votes, more than the 88 needed to win.

These results would then be reported around the country. They were printed in Richmond, Virginia, just three days later, for example. As had occurred eight years earlier, the results reached major population centers and along the coasts by the end of the first week of January. Anybody who was eager to know probably found out by early February, at the latest, unless they lived in a very remote area, or hadn't bothered to go into town in many weeks to find out.

This previous answer of mine in this sub about news of the death of George Washington may also shed some light. In all the major U.S. cities in 1799, the news was being reported within about 3 weeks of the story first being reported in the press. Considering most people didn't live more than two or three weeks away from a major city by riverboat or stagecoach, it's probable that before the telegraph that anybody would hear the latest major news, such as the winner of a Presidential election, within six weeks of a winner being declared, with most Americans hearing the news in less than a month.

But to answer your question more directly on the actual vote count: as detailed above, the votes arrived in D.C. within about a month, though if enough of the vote had arrived before then that put a candidate over the top, then a winner could be declared earlier. But technically speaking, the official counting doesn't occur until early in the new year. Same as today. The counting is done within about an hour's time in a joint session of Congress, weeks after the votes are delivered.