A thing to keep in mind is that around 1820, slavery wasn’t nearly as central to most US political disputes as it would eventually be around 1860. There’s four decades in-between. Basically two generations. Sure, some people who would play big roles in the general run-up to the Civil War were already active in politics back in the day, but mostly, the guys who were still mostly ‘young guard’ politicians in 1820 were already retired when the war actually broke out. In fact, most of them were dead by then.
By 1860, one’s position on slavery would ultimately tend to over-ride all other issues. But around 1820, slavery was just one (albeit prominent) issue. Northern Democrats clearly favoured free soil policies even in 1820, but they were still Democrats, and supported the overall Democratic platform. And Monroe was a fairly moderate Democrat; willing to compromise. With the Federalists imploded, the Democrats had become the Big Tent party. The very biggest of tents, in fact. And Monroe was enough of a centrist to get support from the party’s various wings.
Should Monroe veto the Compromise, he’ll surely lose some of that support, up in the free-soil North. But most of the Northern Democrats will stay loyal and vote Democrat. Because... what’s the alternative? Any Democratic defectors are going to be a small minority. To even stand any kind of chance, they’d have to merge with the Federalist remnants. That means the resulting “Northern” party is going to be a blend of Northern Democrat policies and Federalist/Nationalist ones. You might call this a “moderate (neo-)Federalist” party. They’ll probably be called the National Republicans, and their candidate will probably be John Quincy Adams.
This election will have a very different context than the OTL one of 1824, though. At this stage, “National Republicanism” is going to be unpalatable to most of the electorate. Monroe was popular for a reason! Federalism was dead at this point, and Nationalism was not yet well-formed. To wit: in OTL 1824, J.Q. Adams ran as a fairly moderate Democrat, and he didn’t even win the plurality. He only became President thanks to extensive political dealing (the infamous “corrupt bargain”). Only by 1828 did he run unabashedly as National Republican... and he promptly lost. Because even then, the majority of Americans just didn’t support that kind of thing yet. That tells us a lot about our ATL 1820 election! This was the era of Democratic supremacy, and that wasn’t because the rivals had withered away. It’s the other way around: Federalism had withered because the national mood had shifted. Sure, it would later shift again, but in 1820, any contest between a Democrat and a quasi-Federalist was going to be a shoe-in for the Democrat. So what if the Democrat was pro-slavery? Most free soil Democrats would just hold their noses and vote for him anyway.
Just look at the electoral maps, going all the way back to the Jeffersonian revolution of 1800. The trend shows us how Federalism just melts away like snow in sunlight. It melts all the way back to its North-Eastern heartland... and then it starts dying even there. Sure, the North-Western electorate might have tended towards free soil, but they voted Democrat as soon as they got voting rights, and they kept voting it for a decent number of elections. They may have been Northern Jeffersonians, but they sure were Jeffersonians. Just look at some more electoral maps: the first time a free-soiler wins any of the Western states is in 1836. Clearly, the Northern Democrats didn’t care that much about the matter. Not enough to over-ride their broad support for the Democratic platform as a whole.
If a National Republican party emerges to challenge Monroe in 1820, after he vetoes the Compromise, that new party is going to be an almost exclusively North-Eastern club. They’re going to carry New York, New Jersey, and everything North-East from there. Pennsylvania and Delaware will be the swing states. If they win both of those, they’re going to win. But they’re not going to win both of those. In fact, I expect they’ll carry neither one. Pennsylvania’s electorate was surely in favour of free-soil, but the state was also solidly Democratic. The last time Pennsylvania had produced any non-Democratic Electors was in 1800. At best, the National Republicans get a minority of the state’s 24 electors. Meanwhile, Delaware actually went Federalist right through the 1816 election... but never voted against the interests of slavery. Since there were relatively few slaves in Delaware, the doctrine of diffusionism was actually a quite attractive prospect: by ultimately “selling slaves West” as new slave states were admitted, Delaware would supposedly be able to just grow out of slavery over time, without having to make any difficult choices. So come the 1820 election, I expect Delaware to vote Monroe, just as it did in OTL.
So what’s the outcome here? The National Republicans get formed earlier, and Adams runs earlier... but he loses. Instead of an internal split of the Democrats into multiple wings, it’s more of a clean break into two factions, with a defined geographical boundary. Both factions are going to close ranks immediately, which means Adams (or anyone replacing him) isn’t going to win in ’24, either. Or in ’28, for that matter. The Democrats won’t be making deals with him: they’ll be united in opposing the National Republicans, and they’ll have a majority. Their leading man is probably still Jackson, and he’s going to be President in ’24, just like he wanted in OTL.
The partisanship of this ATL will make the Western states pretty staunchly Democratic, but they would have economic reasons to eventually come around to certain policies favoured by the National Republicans— as happened in OTL. And just as in OTL, the man to keep the West out of the North-East’s orbit would obviously be Henry Clay. He’s the man of the West, and he has his own brand of Nationalism. In the ATL, he wouldn’t call it that, obviously. And the realities of the political landscape would preclude any bargain with the National Republicans. Instead, he’d make his deals with the incumbent Jackson, who would want to get Clay fully on-board to secure the North-West.
So... Jackson/Clay ’28? I could see it happen. And then Clay would get his shot in ’32, with Jackson’s backing. Naturally, he’d need a solid Southerner as Vice President, to balance the ticket. And who’s more of a solid Southerner than John C. Calhoun?
By that point, the National Republicans would be pretty fed-up. They keep winning the same states over and over again, but they just can’t break out of the North-East. Jefferson’s political heirs have been in charge for over three decades! There’s been exactly one Northern President in 48 years— and he left office 32 years ago! Now, Clay’s canny politicking has ensured that Pennsylvania and the North-West have remained firmly in the Democratic camp. In the end, the Democratic offer to these free states (“you don’t talk about slavery, and we’ll just give you whatever you want otherwise... which happens to be pretty in-line with what we want anyway”) is just too good to overcome. So from a New England perspective, it looks like every future election is going to belong to the Democrats... just like nearly every past election.
So, instead of OTL’s Nullification Crisis, a Clay/Calhoun victory in ’32 may just spark the North-Eastern Secession. And I don’t think there’s going to be a majority in favour of trying to wrangle the seceded states back into the Union. After all, Southerners have maps, too. Even without the Monroe doctrine, most of the Western territories are too far North to credibly become slave states. And the Texian Revolution hasn’t happened yet... So letting the North-East secede actually looks like an easy way to definitively ensure that the slave states will always retain their majority.
Without the common foe to keep them united, we’ll probably see an acrimonious split between Clay and Calhoun (and sooner rather than later), followed by competing Presidential bids in ’36. Calhoun almost certainly wins that one, in a rump-USA dominated by slave states. The Democrats get split in twain, with a Northern free soil party and a Southern slavery party. I don’t see the Northerners joining the earlier secessionists right away, but President Calhoun is going to be pushing for the annexation of Texas (and for it joining the Union as multiple slave states, I expect). Soon enough, the free states will realise that they’ll always be a minority in the Union.
Eventually, there’s going to be a Second Northern Secession. Depending on how things develop, this might happen as early as 1840. Probably later, though. I don’t think this reduced USA could easily take as much of Mexico as in OTL, but a “Texas War” in the 1840s looks very credible. The free states in the North would see it as a waste, but the South would go full jingo. Afterwards, with more slave states added (carved out of Texas), I think the disaffected North would start edging towards the exit. Secession by 1850 at the latest, I’d wager. Again, I don’t see the slave power doing much to hinder such a move.
Almost immediately, there will be rapprochement between the newly-independent North-West and already-independent North-East. Possibly confederation. Maybe even full (re-)union.
There wouldn’t be any attempts by the Northern state(s) to subdue the Southern rump-USA. Sure, slavery is a moral problem if it’s happening in your country. But did the Northern states in OTL ever suggest “hey, maybe we could just stop buying those bloodstained raw materials that the South keeps selling to us”? Nope. They talked loudly about the evils of slavery, all while their factories eagerly turned the products of slavery into manufactures that they then sold abroad at a tidy profit. In this ATL, it would be much the same. The Northern state(s) would impose high tariffs on imports from the rump-USA, but high tariffs would be part of their economic policy anyway. Meanwhile, since slavery would no longer be their problem, they’d look the other way. Much as Britain et al. looked the other way where Brazil (or, indeed, the OTL USA) was concerned.
Slavery would have to die a natural death. On the plus side, without decades of digging their heels into the sand, the slavocrats wouldn’t produce the fire-eater mentality. So the relevant OTL example here is, again, Brazil. We should note that every Western nation managed to abolish slavery without having to fight a war over it. The USA was the outlier in OTL, difficulty-wise. Brazil was the outlier timetable-wise. I expect that in this ATL, the resolution of slavery in the USA would end up being a lot like that in Brazil in OTL. It happens later, but it happens without warfare.