If no Confederacy is formed, I'd be surprised if anyone other than South Carolina seceded. At the very least, the Deep South cotton states seceded only because they knew there'd be able to band together. If there is no Confederacy or attempt to join up, not even the other Deep South states will secede. There is zero chance of the Upper South seceding.
If that happens, South Carolina remains very isolated and alone. Lincoln moves very cautiously in handling this one state. He doesn't want to inadvertently provoke the other Deep South states to secede. Lincoln upholds the sanctity of the union, and reaches out to moderates and unionists in the rest of the south to hold the union together. Lincoln is very strict of not allowing the expansion of slavery, but he would support measures maintaining the institution in the states as it currently exists.
Eventually, I think South Carolinia will go back into the fold under ongoing political pressure and some kind of compromise.
Secession will be defeated culturally, but it'll be very vague whether South Carolina "legally" seceded, or if it was actually illegal. It'll remain unproven until the next crisis.
The impact of this is huge. There is no emanicipation. But precedent has been set that the South can survive under a Republican administration. From this point on, the election of a Republican will not set off a secessionist crisis. Instead, it would be specific acts of a Republican administration that might result in a future crisis.
Lincoln is probably too canny to provoke the South, but future Republicans might be more avowedly abolitionist. However, the Republican Party will be able to use the patronage of the Federal government to grow an indigenous Republican Party in the South, probably an alliance of old Southern Whigs and Appalachian Democrats.
A significant Republican Party can probably be built in the border states - Virginia, Tennessee, and Arkansas. There will also be local parties in northern Alabama, western North Carolina, and New Orleans. The question is whether those local parties will have enough time to shift politics to affect things in the next crisis. If by 1864 or 1868, it becomes obvious that the majority of Southerners have reconciled themselves to Republican hegemony and that slavery will slowly die out, and that seceding from the Union is not worthwhile, we will probably see a decades long policy where slavery is more regulated, controlled, and eventually abolished.
On the other hand, if this only delays and not prevents a future secessionist crisis, then a major "learning moment" is that forming a Confederacy is absolutely essential if secession is to succeed. At that point, we have our own Civil War, but one where more states might remain in the Union. It all depends on how local events proceed. We might see all the states of our Confederacy secede, or one where none of the Upper South secedes.