Current evidence suggests that $\omega$ Cen is the nuclear star cluster of a galaxy that merged with the Milky Way at early times. We use Apache Point Observatory Galactic Evolution Experiment (APOGEE), Gaia, Multi Unit Spectroscopic Explorer, and Hubble Space Telescope data supplemented by galaxy chemical evolution models to place constraints on the assembly and chemical enrichment history of $\omega$ Cen. The APOGEE data reveal three stellar populations occupying separate loci on canonical chemical planes. One population resembles metal-poor halo field stars (P1), a second shows light-element abundance anticorrelations typical of metal-poor globular clusters (IM), and a third population (P2) is characterized by an extreme 'second-generation' abundance pattern. Both P1 and P2 populations cover a broad range of metallicity, consistent with extended histories of bursty star formation (SF), which is also evident from their light and $\alpha$-element abundance patterns. Conversely, the IM stars exhibit a narrow metallicity spread, combined with Al─Mg, Na─O, and C─N anticorrelations resembling metal-poor Galactic globular clusters. Moreover, these three populations alone seem to account for the distribution of $\omega$ Cen stars in the chromosome map. We discuss these findings in the context of a scenario according to which $\omega$ Cen formed by a combination of in situ SF within the host galaxy (P1), followed by the spiralling in of gas-rich globular clusters (IM), leading to another burst of SF (P2). We perform a robust comparison of the chemical composition of $\omega$ Cen with those of halo substructures well represented in APOGEE DR17, finding no chemical associations to a high confidence level.