Andrew Jackson dies, tripping on the stairs or something, some short while before the nullification crisis escalates, before the Tariff of 1832 replaces the one of 1828, and before Calhoun resigns the vice presidency. Calhoun becomes president, refuses to accept the proposed Tariff of 1832 as a compromise, and openly supports nullification-- as president. South Carolina nullifies the Tariff of 1828. Utter chaos ensues, Calhoun refuses to act against South Carolina, and in fact openly supports the state's efforts. Various parties attempt to claim that the vice president, in a case such as this one, is merely acting as president, and should not have full powers.
The coming presidential election will be utterly contentious. Calhoun is immediately nominated by a revived Nullifier Party (or States Rights Party, or whatever they wish to call themselves). Northern opponents of nullification (basically: the National Republican Party) have already nominated Henry Clay, who swears that if elected, he will bring South Carolina back into the fold "by force of arms, if needed". The Democrats are divided. Many southern Democrats want to nominate Calhoun as their candidate. Northern Democrats are less keen on that idea. Their convention proves irreconcilable, leading to a schism. The "Northern Democratic Party" nominates Martin Van Buren, while the main Democratic Party (consisting mostly of the southern wing of the party) nominates Calhoun, who vows to uphold the rights of "these united states, even against the ambitions of the general government itself".
Opponents of nullification have lost willingness to compromise on the tariff by this point. It'll be a fight to the end, now.
In any case, in the escalating turmoil, South Catolina makes it clear that if its "sacred rights" are violated (that is: if anyone other than Calhoun wins the coming election), the state will secede from the Union. Other southern states, although far from as radicalised as South Carolina, view the idea of a hypothetical president Clay sending the army against South Carolina as intolerable. If Clay wins, the Union may well tear itself apart. But many in the north consider Calhoun a traitor to the Union already. If he wins, the effect will be much the same.
As the election draws near, nullification might well turn into secession... and war.