Spin excitations play a fundamental role in understanding magnetic properties of materials, and have significant technological implications for magnonic devices. However, accurately modeling these in transition-metal and rare-earth compounds remains a formidable challenge. Here, we present a fully first-principles approach for calculating spin-wave spectra based on time-dependent (TD) density-functional perturbation theory (DFPT), using nonempirical Hubbard functionals. This approach is implemented in a general noncollinear formulation, enabling the study of magnons in both collinear and noncollinear magnetic systems. Unlike methods that rely on empirical Hubbard U parameters to describe the ground state, and Heisenberg Hamiltonians for describing magnetic excitations, the methodology developed here probes directly the dynamical spin susceptibility (efficiently evaluated with TDDFPT throught the Liouville-Lanczos approach), and treats the linear variation of the Hubbard augmentation (in itself calculated non-empirically) in full at a self-consistent level. Furthermore, the method satisfies the Goldstone condition without requiring empirical rescaling of the exchange-correlation kernel or explicit enforcement of sum rules, in contrast to existing state-of-the-art techniques. We benchmark the novel computational scheme on prototypical transition-metal monoxides NiO and MnO, showing remarkable agreement with experiments and highlighting the fundamental role of these newly implemented Hubbard corrections. The method holds great promise for describing collective spin excitations in complex materials containing localized electronic states.