We propose an ab initio Boltzmann transport approach taking into account magnon-phonon scattering (MPS) and three-phonon scattering simultaneously to accurately evaluate the thermal transport properties of ferromagnetic crystals. Using this approach, we studied the nonelectronic thermal transport properties of the body-centered cubic iron as a case. The reasonable agreement between our calculation results and the available experimental data suggests that phonons dominate the nonelectronic thermal conduction at high temperatures, and magnons may contribute to the thermal conductivity only at low temperatures. Remarkably, the abnormal increase in the magnon thermal conductivity at high temperatures implies that other magnon-involved scattering events instead of MPS should dominate the magnon thermal conductivity. Moreover, analyses of average scattering rates and heat propagation lengths suggest that hydrodynamic heat transport may occur at low temperatures. This new approach fills the gap in the first-principles evaluation of the coupled magnon-phonon thermal transport properties in magnetic crystals. Our results will provide valuable references for further investigations of the interplay between magnons and phonons and broaden relevant research prospects about heat management and energy manipulation.