We numerically investigate the dynamic control over the spontaneous emission rate of quantum emitters using tunable hyperbolic metamaterials (HMMs). The dispersion of a metal-dielectric thin-film stack at a given frequency can undergo a topological transition from an elliptical to a hyperbolic dispersion by incorporating a tunable metal or dielectric film in the HMM. This transition modifies the local density of optical states of the emitter and, hence, its emission rate. In the visible range, we use an HMM consisting of TiN and ${{\rm Sb}_2}{{\rm S}_3}$Sb2S3 and show considerable tunability in the Purcell enhancement and quantum efficiency as ${{\rm Sb}_2}{{\rm S}_3}$Sb2S3 phase changes from amorphous to crystalline. Similarly, we show tunable Purcell enhancement in the telecommunication wavelength range using a ${\rm TiN}/{{\rm VO}_2}$TiN/VO2- HMM. Finally, tunable spontaneous emission rate in the mid-IR range is obtained using a ${\rm graphene}/{\rm MgF}_2$graphene/MgF2 HMM by modifying the graphene conductivity through changing its chemical potential. We show that using a metal nitride (for the visible and NIR HMMs) and a fluoride (for the mid-IR HMM) is important to get an appreciable change in the effective permittivity of the thin-film multilayer stack.