What determines the nuclear radio emission in local galaxies? We combine optical [O III] line emission, robust black hole (BH) mass estimates, and high-resolution e-MERLIN 1.5-GHz data, from the LeMMINGs survey, of a statistically-complete sample of 280 nearby, optically active (LINER and Seyfert) and inactive HII and Absorption line galaxies [ALG]) galaxies. Using [O III] luminosity (L[O III]) as a proxy for the accretion power, local galaxies follow distinct sequences in the optical-radio planes of BH activity, which suggest different origins of the nuclear radio emission for the optical classes. The 1.5-GHz radio luminosity of their parsec-scale cores (Lcore) is found to scale with BH mass (MBH) and [O~III] luminosity. Below MBH∼10**6.5 M⊙, stellar processes from non-jetted HII galaxies dominate with Lcore∝M**0.61±0.33BH and Lcore∝L**0.79±0.30[O III]. Above MBH∼10**6.5 M⊙, accretion-driven processes dominate with Lcore∝M**1.5−1.65BH and Lcore∝L**0.99−1.31[O III] for active galaxies: radio-quiet/loud LINERs, Seyferts and jetted HII galaxies always display (although low) signatures of radio-emitting BH activity, with L1.5GHz≳10**19.8 W Hz−1 and MBH≳10**7 M⊙, on a broad range of Eddington-scaled accretion rates (m˙). Radio-quiet and radio-loud LINERs are powered by low-m˙ discs launching sub-relativistic and relativistic jets, respectively. Low-power slow jets and disc/corona winds from moderately high to high-m˙ discs account for the compact and edge-brightened jets of Seyferts, respectively. Jetted HII galaxies may host weakly active BHs. Fuel-starved BHs and recurrent activity account for ALG properties.