  [Abridged] The relation between nuclear star clusters (NSCs) and the growth
of the central SMBHs, as well as their connection to the properties of the host
galaxies, is crucial for understanding the evolution of galaxies. Recent
observations have revealed that about 10 per cent of nucleated galaxies host
hybrid nuclei, consisting of both NSCs and accreting SMBHs that power active
galactic nuclei (AGN). Motivated by the potential of the recently published
multi-wavelength data sets from LeMMINGs survey, here we present the most
thorough investigation to date of the incidence of hybrid nuclei in a large
sample of 100 nearby nucleated galaxies (10 E, 25 S0, 63 S, and 2 Irr),
covering a wide range in stellar mass ($M_{*,\rm gal} \sim 10^{8.7}-10^{12}~\rm
M_{sun}$). We identify the nuclei and derive their properties by performing
detailed 1D and 2D multi-component decompositions of the optical and
near-infrared $HST$ stellar light distributions of the galaxies using S\'ersic
and core-S\'ersic models. Our AGN diagnostics are based on homogeneously
derived nuclear 1.5 GHz $e$-MERLIN radio, $Chandra$ X-ray (0.3--10 keV) and
optical emission-line data. We determine the nucleation fraction ($f_{\rm nuc}
$) as the relative incidence of nuclei across the LeMMINGs $HST$ sample and
find $f_{\rm nuc} =~ $100/149 (= 67 $\pm$ 7 per cent), confirming previous
work, with a peak value of 49/56~(= $88 \pm 13$ per cent) at bulge masses
$M_{*,\rm bulge} \sim 10^{9.4}$- $10^{10.8}~\rm M_{sun}$. We identify 30
nucleated LeMMINGs galaxies that are optically active, radio-detected and X-ray
luminous ($L_{X} > 10^{39}$ erg s$^{-1}$). This indicates that our nucleated
sample has a lower limit $\sim$ 30 per cent occupancy of hybrid nuclei, which
is a function of $M_{*,\rm bulge}$ and $M_{*,\rm gal}$. We find that hybrid
nuclei have a number density of $(1.5 \pm 0.4) \times 10^{-5}$ Mpc$^{-3}$.
