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), $L_{\rm R,core}-M_{*, \rm bulge}$ , relations, with a mass turnover at $M_{*, \rm bulge}\sim 10^{9.8 \pm 0.3} \rm { M_{\odot }}$ (supermassive blackhole mass $M_{\rm BH} \sim 10^{6.8 \pm 0.3} \rm M_{\odot }$ ), 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 bulge-less galaxies host an AGN. The AGN fraction increases with increasing $M_{*,\rm bulge}$ such that $f_{\rm optical\_AGN}\propto M_{*,\rm bulge}^{0.24 \pm 0.06}$ and $f_{\rm radio\_AGN}\propto M_{*,\rm bulge}^{0.24 \pm 0.05}$ . Between $M_{*,\rm bulge}\sim 10^{8.5}$ and $10^{11.3} \rm M_{\odot }$ , $f_{\rm optical\_AGN}$ 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 Sérsic 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.