  [Abridged] We use high-resolution $HST$ imaging and $e$-MERLIN 1.5-GHz
observations of galaxy cores from the LeMMINGs survey to investigate the
relation between optical structural properties and nuclear radio emission for a
large sample of galaxies. We perform accurate, multi-component decompositions
of new surface brightness profiles extracted from $HST$ images for 163 LeMMINGs
galaxies and fit up to six galaxy components (e.g., bulges, discs, AGN, bars,
rings, spiral arms, and nuclear star clusters) simultaneously with S\'ersic
and/or core-S\'ersic models. By adding such decomposition data for 10 LeMMINGs
galaxies from our past work, the final sample of 173 nearby galaxies (102 Ss,
42 S0s, 23 Es plus 6 Irr) with bulge stellar mass (typically) M_*, bulge ~
10^6-10^12.5 M_sun, encompasses all optical spectral classes (LINER, Seyfert,
ALG and H II). We show that the bulge mass can be significantly overestimated
in many galaxies when components such as bars, rings and spirals are not
included in the fits. We additionally implement a Monte Carlo method to
determine errors on bulge, disc and other fitted structural parameters. Moving
(in the opposite direction) across the Hubble sequence, i.e., from the
irregular to elliptical galaxies, we confirm that bulges become larger, more
prominent and round. Such bulge dominance is associated with a brighter radio
core luminosity. We also find that the radio detection fraction increases with
bulge mass. At M_*,bulge > 10^11 M_sun, the radio detection fraction is 77%,
declining to 24% for M_bulge < 10^10 M_sun. Furthermore, we observe
core-S\'ersic bulges tend to be systematically round and to possess high radio
core luminosities and boxy-distorted or pure elliptical isophotes.
