We establish the spaser generation conditions for nonlocal plasmonic lasers and nonlocal plasmonic core-shell lasers based on the full-wave nonlocal Mie theory. Numerical results show that the required gain threshold and the gain refractive index become large generally when the nonlocality or spatial dispersion is taken into account. This tendency can be understood by the analysis with the proposed equivalent permittivity for nonlocal metallic nanoparticles. Since the nonlocality for the compact nanoparticles is quite significant, our study will be of great importance in the design of ultrasmall nanoparticle lasers.