  Multiwavelength studies indicate that nuclear activity and bulge properties
are closely related, but the details remain unclear. To study this further, we
combine $Hubble~Space~Telescope$ bulge structural and photometric properties
with 1.5 GHz, $e$-MERLIN nuclear radio continuum data from the LeMMINGs survey
for a large sample of 173 `active' galaxies (LINERs and Seyferts) and
`inactive' galaxies (H IIs and absorption line galaxies, ALGs). Dividing our
sample into active and inactive, they define distinct (radio core
luminosity)$-$(bulge mass), L_R,core-M_*,bulge, relations, with a mass turnover
at M_*, bulge ~ 10^(9.8 +- 0.3) M_sun (supermassive black hole mass M_BH ~
10^(6.8 +- 0.3) M_sun), which marks the transition from AGN-dominated nuclear
radio emission in more massive bulges to that mainly driven by stellar
processes in low-mass bulges. None of our 10/173 bulgeless galaxies host an
AGN. The AGN fraction increases with increasing M_*, bulge such that
f_optical_AGN $\propto$ M_*,bulge^(0.24 +- 0.06) and f_radio_AGN $\propto$
M_*,bulge^(0.24 +- 0.05). Between M_*,bulge ~ 10^8.5 and 10^11.3 M_sun,
f_optical_AGN steadily rises from 15 +- 4 to 80 +- 5 per cent. We find that at
fixed bulge mass, the radio loudness, nuclear radio activity and the (optical
and radio) AGN fraction exhibit no dependence on environment. Radio-loud hosts
preferentially possess an early-type morphology than radio-quiet hosts, the two
types are however indistinguishable in terms of bulge S\'ersic index and
ellipticity, while results on the bulge inner logarithmic profile slope are
inconclusive. We finally discuss the importance of bulge mass in determining
the AGN triggering processes, including potential implications for the nuclear
radio emission in nearby galaxies.
