We suggest a novel class of active nanoantennas based on diamond nanoparticles with embedded nitrogen-vacancy centres coupled to Mie resonances of nanoparticles. We theoretically study the optical properties of such nanoantennas including the field enhancement and Purcell effect, and experimentally demonstrate the enhancement of the fluorescence rate of the emitters due to particle resonances, as compared to a nonresonant regime. Our results pave the way towards active dielectric nanophotonics for quantum light sources, bioimaging, and quantum information processing.