The problems of observability and identifiability have been of great interest as previous steps to estimating parameters and initial conditions of dynamical systems to which some known data (observations) are associated. While most works focus on linear and polynomial/rational systems of ODEs, general nonlinear systems, including non-analytic systems have received far less attention and, to the best of our knowledge, no unified constructive methodology has been proposed to assess and guarantee parameter and state recoverability in this setting. Some symbolic tools provide automated analyses for rational or nonlinear, analytical systems, offering qualitative identifiability and observability verdicts that are sometimes incomplete. In this work, we introduce a family of efficient and fully constructive procedures that can enable explicit recovery of the unknown parameters and/or initial conditions, whenever possible, for a large class, not necessarily rational or analytic, nonlinear ODE systems. Each procedure is tailored to different observational scenarios and based on the resolution of linear systems. As a case study, we apply these procedures to several epidemic models, with a detailed focus on the SIRS model, demonstrating its joint observability-identifiability when only a portion of the infected individuals is measured, a scenario that has not been studied before. In contrast, for the same observations, the SIR model is observable and identifiable, but not jointly observable-identifiable. This distinction allows us to introduce a novel approach to discriminating between different epidemiological models (SIR vs. SIRS) from short-time data. For these two models, we illustrate the theoretical results through some numerical experiments, validating the approach and highlighting its practical applicability to real-world scenarios.