We explore the effect of pulsars, in particular those born with millisecond periods, on their surrounding supernova ejectas. While they spin down, fast-spinning pulsars release their tremendous rotational energy in the form of a relativistic magnetized wind that can affect the dynamics and luminosity of the supernova. We estimate the thermal and non-thermal radiations expected from these specific objects, concentrating at times a few years after the onset of the explosion. We find that the bolometric light curves present a high luminosity plateau (that can reach 10⁴³-10⁴⁴ erg s^-1) over a few years. An equally bright TeV gamma-ray emission, and a milder X-ray peak (of the order of 10⁴⁰-10⁴² erg s^-1) could also appear a few months to a few years after the explosion, as the pulsar wind nebula emerges, depending on the injection parameters. The observations of these signatures by following the emission of a large number of supernovae could have important implications for the understanding of core-collapse supernovae and reveal the nature of the remnant compact object.