Galactic nuclei typically host either a nuclear star cluster (NSC, prevalent in galaxies with masses <~10^10^M_{sun}_) or a massive black hole (MBH, common in galaxies with masses >~10^12^M_{sun}_). In the intermediate-mass range, some nuclei host both an NSC and an MBH. In this paper, we explore scaling relations between NSC mass (M_*,gal_) and host-galaxy total stellar mass (M_NSC_) using a large sample of NSCs in late- and early-type galaxies, including a number of NSCs harbouring an MBH. Such scaling relations reflect the underlying physical mechanisms driving the formation and (co)evolution of these central massive objects. We find ~1.5{sigma} significant differences between NSCs in late- and early-type galaxies in the slopes and offsets of the relations r_eff,NSC_-M_NSC_, r_eff,NSC_-M_*,gal_ and M_NSC_-M_*,gal_, in the sense that (i) NSCs in late types are more compact at fixed M_NSC_ and M_*,gal_; and (ii) the M_NSC_-M_*,gal_ relation is shallower for NSCs in late types than in early types, similar to the M_BH-M_*,bulge relation. We discuss these results in the context of the (possibly ongoing) evolution of NSCs, depending on host-galaxy type. For NSCs with an MBH, we illustrate the possible influence of an MBH on its host NSC, by considering the ratio between the radius of the MBH sphere of influence and r_eff,NSC_. NSCs harbouring a sufficiently massive black hole are likely to exhibit surface brightness profile deviating from a typical King profile.