While it was true that several radicals (Fremont, Chase, Wade, Winter Davis, Pomeroy, etc) were actively campaigning against Lincoln's renomination, much of the party infrastructure rallied behind the incumbent. Flexing his political acumen, Lincoln from the outset of this campaign focused more on securing the backing of the state conventions and the national committee (both of which would be much more crucial in the nominating convention) than the big name politicians working against him (whom he would start to woo after his renomination was secured). Lincoln saw to it that one of his closest allies, Henry J. Raymond, was installed as chairman of the RNC, and the case in point of his state convention strategy came with Ohio. When Lincoln was able to maneuver that body into endorsing his renomination that essentially took all the wind out of the sails of the "Draft Chase" movement. This is all to say that as long as Lincoln is actively seeking the nomination he will likely receive it.
Therefore, in order for Lincoln to be out of the picture he must voluntarily make that choice. This is out of Lincoln's OTL character, but perhaps something happens to dissuade him from a second run. In that case, I think Chase would be the most likely nominee. He was the radical's favorite stalking-horse replacement for Lincoln, so assuming he avoids his OTL humiliation at Lincoln's hands I believe they would likely go forward with him. After Lincoln, Chase probably had the second-most campaign infrastructure set up for the Republican contest as well as experience and reputation for 1860, so he could likely earn the nomination easily enough at the convention. Being a Midwestern former Democrat, Chase would likely seek a Northeastern former Whig to balance the ticket. If he was looking to mend ties with Seward (who would have almost certainly been leading opposition him behind the scenes) and sure up his support in the crucial state of New York, then he might choose Senator Ira Harris. More likely, however, he would attempt to sideline the more moderate/conservative faction of the party under Seward and instead double-down with a radical running-mate. That could be someone like Governor John A. Andrew of Massachusetts or Representative Justin S. Morrill of Vermont.
The ultimate winner of the election would depend, of course, on the course of the Civil War. If the war continues to go badly than McClellan could likely sweep the contest (especially facing a more radical opponent without incumbency). If the war turns for the better than Chase could likely pull off the victory. Assuming the war follows a similar course, Chase wins the election, and the war ends similarly to OTL, Reconstruction could be in for much less contention. ITTL, it wouldn't even matter if the president was assassinated because his running-mate would be Radical as well. This would likely allow for a much more unified Reconstruction policy than occurred IOTL. Instead of the executive and legislative branches squabbling and weakening Reconstruction by the competing projects, this TL would feature a executive and legislative united behind one objective. This would likely result in a more successful Reconstruction overall, but they would need to avoid the true death-kneel of the project, the Long Depression, for it to be truly successful in the long-term.
