Freeing slaves was never in the North's plan before the war started. In 1860, Lincoln and the Republicans had run on a platform of repealing the Kansas-Nebraska Act and repealing the the Fugitive Slave Act. They had no intention of interfering with slavery in states where it already existed.

The repeal of the Kansas-Nebraska Act would ensure that the unorganized territories wouldn't become slave states--the settlers there overwhelmingly supported being free states (being slave states meant that laborers would be competing with slaves for jobs, and would drive the price of farmers' crops down since slave plantations could produce the same crops at a cheaper price). As territories, they were already all free territories, and they had no desire to change course once they were admitted into the Union as states. The Kansas-Nebraska Act, however, made it so that pro-slave forces would almost certainly be able to get at least some of the territories admitted as new slave states, despite the popular will of the people who lived in those territories.

The Fugitive Slave Act of 1850 had formed a federal force of U.S. Marshals who were responsible for retrieving escaped slaves who made it to Northern states. Because of the Dred Scott decision by the Supreme Court in 1857, slaves had no federal rights and so they had no legal recourse to defend themselves against U.S. Marshals who came to take them, and this was a real problem, because a lot of times, free black Northerners would be accused of being escaped slaves that they weren't. But they weren't allowed to try to prove otherwise in federal court due to the Dred Scott decision. The North wanted to defend the rights of their black citizens, and they didn't want to live under a federal government that was enforcing slave laws in free states. So they wanted to repeal the law, and at least go back to the original law, which was the Fugitive Slave Act of 1793. Under that law, the slave-owners themselves, or else their hired bounty hunters, had to come North to retrieve escaped slaves, and if there was a dispute, then the accused slave could file a lawsuit in state court, where black citizens did have standing to prove that they might not be the slave they were accused of being.

Anything beyond repealing the Kansas-Nebraska Act and the Fugitive Slave Act of 1850 was not really part of the immediate dispute that led to Lincoln and the Republicans getting elected in 1860 which caused the South to secede. In general, Republicans did support emancipation, but the vast majority supported a plan--at some point in the future--of gradual emancipation. Lincoln and others envisioned a situation as had happened in New York and New Jersey, where slavery had been phased out over several decades. Both those states passed their emancipation laws before 1800, but they went into effect over time, with the slaves in New York being freed in 1827, and in New Jersey in 1846.

But that was still off in the future. More immediately, they envisioned the Western territories becoming free states once the Kansas-Nebraska Act was repealed, and that would eventually, in some 50-odd years time, give them enough votes in the Senate to begin enacting a gradual emancipation plan, once those territories became free states and each had two Senators. (The Senate was appointed back then by state legislatures, and were not elected, so if a state became a slave state, it was virtually guaranteed that any Senator appointed to the U.S. Senate would be pro-slavery.)

Once they had the votes in both Houses of Congress, then and only then would emancipation be discussed. And it would likely be discussed along the terms that New York and New Jersey had abolished it--that is, give slave-owners some 30-50 years to prepare, by figuring out how they would start paying the former slaves a small wage, and by allowing them time to divest their finances from slave investments.

But again, that was well off into the future. The South seceded because they didn't want anything like that to happen, ever, and just the threat of going back on the pro-slavery progress they had made in the 1850s when they got the Fugitive Slave Act passed (which replaced the earlier, more equitable Fugitive Slave Act of 1793) and the Kansas-Nebraska Act passed (which replaced the earlier, more equitable Missouri Compromise of 1820) was too much of a threat to long-term viability of slavery that they decided they were better off seceding than living under the U.S. Constitution any longer. With Lincoln in the White House and the Republicans controlling the House (though the Democrats still controlled the Senate by a single vote), it was very likely they were going to have to give up some parts of their recent political wins, and they did not want to do that. So they seceded. A Peace Conference was held in February 1861, the month before Lincoln took office, but when it went nowhere, the South dug in their heels, and the war broke out barely a month after Lincoln was sworn in as President.

The reason emancipation moved forward during the war was that the political calculus changed due to that war. Lincoln issued the Emancipation Proclamation freeing the slaves only in states that had seceded in an effort to turn popular opinion in those states against their Confederate governments and against the war. It was a message: If you play nice with Constitution like the slave states of Kentucky, Maryland, and Delaware, you're going to get to keep your slaves, so you better overthrow your Confederate leadership if you want to keep your slaves, too. It was simultaneously a message to Lincoln's base of support in the North who thought he was being too agreeable to the South, and he wanted to prove himself among the Republican Party, who had urged him to be more aggressive.

Then, the 13th Amendment moved forward in early 1865 because, by that point, the war was basically won, but there was no way that the people of the North were going to accept a peace under which the South could come back under the Constitution with their slaves intact as if nothing had ever happened. So Republicans in Congress began moving forward with the plan to abolish slavery for good.

But had the South never seceded, and had the Democrats been willing to work with the Republicans in Congress, then the likely outcome of Lincoln's presidency would have been some kind of repeal both the Kansas-Nebraska Act and the Fugitive Slave Act that put the country back to about where it was before the presidency of Millard Fillmore. In effect, in about 30-50 years, as all the rest of the continental territory got filled in as new states, the abolitionists would have reached enough votes in the Senate to seriously start talking about gradual emancipation on the national level. And 30-50 years after that, gradual emancipation would have been complete. So the last slaves would have been freed sometime between the 1920s and 1960s instead of in the 1860s. There were advocates in the 1860s for immediate emancipation, but that was largely a fringe opinion. The popular opinion of the Republican Party was some kind of plan of gradual emancipation, but that was going to have to wait until sometime after the repeal of the Kansas-Nebraska Act and the Fugitive Slave Act, and they knew it, and were willing to work from that starting point.