Isaac Newton Arnold, in his 1866 biography The History of Abraham Lincoln, notes on page 154:
On the 7th of November, 1860, it was known throughout the Republic that Lincoln was elected. He could not be inaugurated until the 4th of march following. For these four eventful months the conspirators had control of the Federal government.
November 7th is the day following the election, held on November 6, 1860.
So by the 1860 election, with the intercontinental telegraph still only about 3/4 complete but the Pony Express in operation covering the gap, election results were known the day after the election. Note further that this was a close election, hinging on Lincoln's victory of New York State's 35 electoral college votes by a margin of just 53.7% to 46.3%. Without those 35 electoral college votes going to Lincoln he was 7 short of election, and determination of the presidency would have gone to a run-off vote in the House of Representatives.
In regards an earlier Presidential election, that in 1828 of Andrew Jackson, John S. Jenkins notes on page 176 of his 1847 biography The Life of General Andrew Jackson:
Not long after the result of the  election, General Jackson experienced a most afflicting bereavement, in the death of his amiable wife [on December 22, 1828].
Note that this (1828) Presidential election is occurring late in the transition, from the doubly indirect process of having State Legislatures select the Electoral College electors, to the singly indirect one of having a popular vote to determine State electors to the Electoral College. In 1828 specifically the Presidential Election runs from October 31 to December 2, with just Delaware and South Carolina still not yet having a popular vote for President and Maryland, Tennessee, Maine, and New York running hybrid selection mechanisms for the Electoral College. In particular, the final two electors from New York State were selected by the other (just elected) 34 electors; requiring first that they actually meet to do so.
Despite this complex and drawn out selection mechanism, and the absence of widespread telegraph service, we have attestation that the election results were promptly known nationally, though the exact time span remains unidentified.