We analyze new high-resolution (400 pc) ∼220 GHz continuum and CO(2-1) ALMA observations of a representative sample of 22 local (z<0.165) ULIRG systems (32 individual nuclei) as part of the "Physics of ULIRGs with MUSE and ALMA" (PUMA) project. The deconvolved half-light radii of the ∼220 GHz continuum sources, r cont , are between <60 pc and 350 pc (median 80-100 pc). We associate these regions with the regions emitting the bulk of the infrared luminosity (L IR ). The good agreement, within a factor of 2, between the observed ∼220 GHz fluxes and the extrapolation of the infrared gray-body, and the small contributions from synchrotron and free-free emission support this assumption. The cold molecular gas emission sizes are between 60 and 700 pc and are, on average, ∼2.6 times larger than the continuum. Using these measurements, we derive the nuclear L IR and cold molecular gas surface densities (Σ L IR = 10 11.5 − 10 14.3 L kpc −2 and Σ H 2 = 10 2.9 − 10 4.2 M pc −2 , respectively). Assuming that the L IR is produced by star-formation, the median Σ L IR corresponds to Σ SFR = 2500 M yr −1 kpc −2 . This Σ SFR implies extremely short depletion times, Σ H 2 /Σ SFR <1-15 Myr, and unphysical star-formation efficiencies >1 for 70% of the sample. Therefore, this favors the presence of an obscured AGN in these objects that could dominate the L IR . We also classify the ULIRG nuclei in two groups: (a) compact nuclei (r cont <130 pc) with high mid-IR excess emission (∆L 6−20µm /L IR ) found in optically classified AGN; and (b) nuclei following a relation with decreasing ∆L 6−20µm /L IR for decreasing r cont . The majority, 65%, of the nuclei in interacting systems lie in the low-r cont end (<130 pc) of this relation, while only 25% of the mergers do so. This suggests that in the early stages of the interaction, the activity occurs in a very compact and dust-obscured region while, in more advanced merger stages, the activity is more extended, unless an optically detected AGN is present. Approximately two thirds of the nuclei have nuclear radiation pressures above the Eddington limit. This is consistent with the ubiquitous detection of massive outflows in local ULIRGs and supports the importance of the radiation pressure in the outflow launching process.