The immediate effect, ignoring the considerable effects on Missouri itself, is that Maine does not get statehood. They want to pair admission of free states with slave states at that point, right? Also no 1820 compromise.
So in 1836-7, Arkansas and Michigan come in, then in 1845-6 Florida, Texas, Wisconsin, and Iowa. Then in 1850 you have the controversy over the admission of California. However much people in Maine may want it, there will be no support among southern politicians for creating yet another state in New England at that point.
But Maine may be admitted as a state anyway, as one of the Republican nineteenth century gerrymandering states, along with West Virginia, Nevada, North Dakota, Colorado, and Wyoming, to boost the number of Republicans in Congress and Republican EVs. The Republicans were prepared to count the electoral votes of occupied Louisiana and Tennessee to secure Lincoln's re-election, so I'm sure they would have been happy to split off Maine from Massachusetts to get Lincoln another two EVs.
This assumes that the Civil War is butterflied away due to no "bleeding Kansas", and as stated above the POD gets interesting at this point. The idea of Kansas as a slave state was pretty insane even given the standards of the time, but it probably would be too much even for the fire-eaters if it didn't even border on any slave states. Douglas still wants his railroad, though. Southern politicians for reasons I never completely understood are bent on expanding slavery.
One possibility is that nothing happens. Douglas has nothing to bargain with, so southern politicians block the railroad on general principles. There is no "bleeding Kansas"controversy. This probably means no one pays attention to Douglas being re-elected to the Senate in Illinois. There is probably still a Republican Party, because of the Fugitive Slave Law but its a lot weaker. If Douglas' presidential campaign is not butterflied way entirely, southern Democrats do not object, though the idea that the split caused the Republicans to win is a historical myth. You probably have Seward losing narrowly to Douglas in the electoral college, and by a wider margin in the popular vote, in 1860.
(maybe by 43% to 38% in the popular vote, with the rest going to Bell. Seward wins the OTL Lincoln states, except for California, Oregon, Illinois, Indiana, and Ohio, so he gets 126 EV compared to 180 for Lincoln, though Ohio and its 27 EV would be winnable for a Seward/ Chase ticket and that would be enough to get Seward over the top. Bell takes Kentucky and Tennessee for 24 EV, with Douglas scraping through with the other 153 EV, though note that Bell wins in any of Maryland, Virginia, or North Carolina, all quite achievable, push the election to the House)
The other possibility is that the fire-eater push things way to far, as they did IOTL, but just find another place than Kansas to do it. In addition to Cuba as stated above, Arizona, New Mexico, and a project to split off southern California and introduce slavery there -which is crazy but no crazier than the Kansas idea- are possibilities.
Dredd Scott either lives his entire life as a free man in this scenario or is not held as a slave in Missouri. This raises similar issues to having no bleeding Kansas. Does the Supreme Court leave well enough alone, or do they find another case to use to butt in on the debate?