Pretty straightforward and he was the runner up at the convention.
More precisely, he was the overwhelming favorite going into the convention. On the evening of May 17, after the platform had been presented, there was a motion to proceed to nominations. But the tally sheets hadn't arrived from the printer. Rather than wait for them, the convention adjourned. During the night of May 17-18, Lincoln's managers made the deal with Simon Cameron that brought Lincoln 56 Pennsylvania votes on the second ballot, which put him into a near tie with Seward and started the bandwagon rolling.
Bruce Catton in
The Coming Fury suggested that if the tally sheets had been available the day before, Seward would have been nominated.
Assume that. Who would be Seward's running mate. Probably not Lincoln, as he was an ex-Whig like Seward, and there was a need to balance the ticket with an ex-Democrat (such as Hamlin was OTL). But also a Westerner. Perhaps Senator Lyman Trumbull of Illinois. (If Seward-Trumbull wins, that will leave Trumbull's Senate seat vacant in 1861, as he was up for re-election. I could see Lincoln being elected to replace him. One assumes Lincoln would swallow his disappointment and stump hard for the Republican ticket; and he had stepped aside for Trumbull in the Senate election of 1854.) Trumbull would offset Douglas' home state advantage in Illinois.
How does it affect the election? Seward wins, I think. As noted elsewhere on the thread, Lincoln won 11 electoral votes by pluralities: 3 in Oregon, 4 in California, and 4 in New Jersey (the last due to a split among Democrats over the Douglas-Bell fusion ticket), but had absolute majorities in 15 states with 169 electoral votes, needing 152 to win. Seward will not do significantly worse in those 15 states. Even losing Illinois (11 EV) wouldn't stop him.
The reaction of the Deep South would be about the same: South Carolina declaring secession in December, with the rest of the "Gulf Squadron" following in January 1861. Assuming no butterfly effects on events...
Seward had some goofy ideas, like trying to provoke a war over Spain's recolonization of the Dominican Republic, in the hope that Southerners would "rally round the flag". But I don't think he'd actually do that.
The big first change is that Seward was prepared to evacuate Fort Sumter, and thus delay the armed confrontation with the secessionists. However, he would not evacuate Fort Pickens, in Florida, which posed the same issue, but under much more favorable conditions for the Union. Fort Pickens, unlike Fort Sumter, cannot be cut off from the sea by batteries in the surrounding harbor; it sits on a bar between Pensacola Bay and the open sea. It would be unassailable by Confederate forces.
So a stalemate could arise. Seward, like Lincoln, would want to avoid a war to restore the Union, and hope for the seceding states to return to the Union voluntarily. He would take no forceful action against the secession-declaring states. The remaining slave states would wait, as it OTL. In the meantime...
At some point, Seward as President would begin to fill the many Federal offices at his disposal: US District Attorneys, US Marshals, postmasters, excisemen and customs officers. His appointments for the free states would be accepted normally. Appointments for the Border States and Upper South would be trickier; but as these states had not declared secession, it would clearly be within his power, and arguably his official duty, to make these appointments. If he was clever, and Seward
was clever, he would find respected local men to appoint (but not abolitionists), gradually "normalizing" his position as President.
This will annoy Jeff Davis and his crew no end. But attacking Fort Pickens is impractical. What would they do, eventually?
Ex-Governor Henry Wise in Virginia will still conspire to seize the Harpers Ferry Arsenal and Norfolk Navy Yard with Virginia militia (though without any official authorization from Governor Letcher or the commanders of the militia).
Missouri governor Claiborne Jackson will still conspire with Davis to achieve secession by coup d'état. Davis, frustrated elsewhere, might push this.
If either of these plots "goes hot"...
OTL, Wise's plot was pre-empted by the bombardment of Fort Sumter and Virginia secession in April 1861 (it was ready to go in a few more days).
OTL, Jackson was thwarted by pre-emptive Unionist action in May 1861.
ITTL, as the stalemate continues...
Probably, Wise's plot goes hot first. The questions are:
Does it succeed? I have no idea about that.
What are the reactions? ISTM that Wise was overreaching. His action would embarrass Letcher and the Virginia government, and would be repudiated. It could seriously discredit the secessionists in Virginia, especially if blood is shed. Even if the plot succeeded without bloodshed, I think Letcher would feel compelled to repudiate the action, and order the militia to withdraw.
Seward could respond with an olive branch of sorts. This would still be early and few Federal appointments would have been made yet; none in Virginia. Seward could propose to Letcher that Federal offices in Virginia be filled with reliable Unionists subject to approval by Letcher and the state's leading ex-Whigs. This could tip the political balance in Virginia back toward Unionism.
Two other processes in train:
Seward may eventually deploy US ships to hover off Southern ports and collect customs duties.
Davis will try to force a fight somewhere.
Which happens first? Idunno. The longer before shooting starts, the greater the chance that the Upper South states don't declare secession. It also depends on how egregious Confederate actions are.
Another question is whether Seward, during the waiting period, seeks to build up Federal forces - just in case. He won't "call for militia" from the states for Federal service, as Lincoln did - that would probably provoke more secession declarations. But more regulars would be safe. However, that might have to wait until Congress meets...
I can't see any more to discuss usefully.