As feedback isn't bad on the plausibility of the idea I'll start on a timeline.
POD: As the nominations for the Constitutional Union's convention approaches in early May 1860 John Bell, whom most assume will take the nomination, meets with Northern Democrat candidate Stephan A. Douglas, nominated in April. Bell comes to Douglas with a proposition which he believes will 'save the union.' He urges Douglas to enter the Constitutional Union's convention seeking the nomination. Bell promises that if Douglas does so he will support him and Douglas promises him the vice-presidential spot. At the convention, after several ballots, Douglas is chosen as their candidate with Bell as his new Vice-presidential candidate.
- Later in May Douglas and Bell come forward to announce that their two parties would be entirely joining together to form the 'Democratic Union Party.' Their chances seem good.
- The election comes around with Breckenridge's Southern Democratic Party strong, as could be expected, in the South. Much of the North, though, seems a toss up between Douglas and Lincoln, who had pitted off in a senate race not to long ago. Finally the results come in and the electoral college (152 needed) stands as such:
Lincoln
- Connecticut (6)
- Illinois (11)
- Indiana (13)
- Iowa (4)
- Maine (8)
- Massachusetts (13)
- Minnesota (4)
- New Hampshire (5)
- Ohio (23)
- Pennsylvania (27)
- Rhode Island (4)
- Vermont (5)
- Wisconsin (5)
Total: 128
Douglas
- California (4)
- Kentucky (12)
- Maryland (8)
- Missouri (9)
- New Jersey (7)
- New York (35)
- Oregon (3)
- Tennessee (12)
- Virginia (15)
Total: 105
Breckenridge
- Alabama (9)
- Arkansas (4)
- Delaware (3)
- Florida (3)
- Georgia (10)
- Louisiana (6)
- Mississippi (7)
- North Carolina (10)
- South Carolina (8)
- Texas (4)
Total: 64
- The election now heads to congress, where the winner will be determined by the 36th US congress (the congress elected in 1858). The House is key to the election as they decide the president. The voting takes place on a state by state basis with 17 votes needed to win. Alabama, Arkansas, Delaware, Florida, Georgia, Louisiana, Mississippi, North Carolina, South Carolina, Texas, Virginia, Kentucky, Tennessee, Maryland, Missouri, Illinois, New Jersey, Oregon and California all back Douglas, giving him 19 votes and the presidency
- In the senate, where the democrats hold a majority already, Bell is easily elected as the Vice-President.
- On March 4th Stephan A. Douglas succeeds Buchanan and is inaugurated as the sixteenth president of the United States of America. In Congress the Republicans now control 130 of the 233 seats, giving them the slim majority. In the senate however the 68 seats are divided between 28 Republicans, Democratic Union 25, Democrats 15, keeping the Republicans out of the majority but still the largest party.
- The Republicans, in early April, present the 'Slave Future Act' which would require a nationwide referendum to be held on the issue of slavery with the victorious side becoming nationwide law. The act easily passes through the Republican controlled house and heads to the house where the Republicans need 6 senators to vote across the aisle with them to pass the act. They succeed in getting exactly that and they pass the act. Douglas, though, refuses to sign the bill and it is vetoed and soon dies in congress where it is far from being able to gain the necessary 2/3 to override the veto.
- This sets the mood for the entire term. Both sides refuse to budge on slavery and essentially refuse to debate anything else. This brings the nations politics to a fever pitch in preparation for the 1862 congressional election. In the run up to the election both the Southern Democrats and the Democratic Union drops a word out of their name, the Southern Democratic Party becoming simply the Southern Party, and the Democratic Union Party becoming the Union Party.
- In the run up to the election the Southern Party and the Union Party also declare a coalition, which they believe will bring them a majority in both houses. The Southern Party agrees to run candidates only in the congressional districts which their candidate won a majority of votes in the previous presidential election and the Union Party will run candidates everywhere else but there.
- This strategy is effective and, for the 38th congress, 149 of the 233 seats in the house and 43 of the 68 senate seats are won by the two parties combined, leaving the Republicans solidly in the minority. This begins an era of coalition rule between the Southern and Union parties which puts them in power in Washington for several years.