We present 5 GHz e-MERLIN radio images at 50 mas resolution of the nuclear regions of the Legacy e-MERLIN Multi-band Imaging of Nearby Galaxies survey (LeMMINGs): the deepest statistically complete radio-band survey of the local Universe ($\lt$120 Mpc), consisting of 280 galaxies spanning all morphological and nuclear types. We detect nuclear radio emission above a median $5\sigma$ threshold of 0.33 mJy beam$^{-1}$ in 68/280 sources (24 per cent), with core luminosities $10^{35}$─$10^{41.9}$ erg s$^{-1}$. The radio emission is attributed to active galactic nuclei (AGNs), circumnuclear star formation, or ─ in the case of NGC 3690 ─ a tidal disruption event. The brightest radio nuclei, with brightness temperatures $\ge 10^{6}$ K, reside in optically 'active' galaxies ─ LINERs and Seyferts. The detection rate for 'inactive' systems (H II and absorption-line galaxies), which may host low-luminosity AGNs (LLAGNs), is 8 per cent. Most detections (78 per cent) are compact ($\lt$10 pc), while the remaining 22 per cent show extended jet-like features (up to 380 pc). Compared to 1.5 GHz LeMMINGs data, the 5 GHz observations' superior resolution and spatial filtering resolve out large-scale structures, isolating genuine nuclear emission. Our results suggest that LLAGNs are the primary manifestation of black-hole activity in the local Universe in the form of compact jets and cores, with a preference for early-type hosts. The two LeMMINGs campaigns show that up to 30 per cent of the local galaxy population host a radio-active nucleus, highlighting the necessity of high-resolution high-sensitivity imaging for uncovering nuclear emission at the lowest luminosities.