Heat transport in bulk materials is well described using the Debye theory of three-dimensional vibrational modes (phonons) and the acoustic match model. However, in cryogenic nanodevices, phonon wavelengths exceed device dimensions, leading to confinement effects that standard models fail to address. With the growing application of low-temperature devices in communication, sensing, and quantum technologies, there is an urgent need for models that accurately describe heat transport under confinement. We introduce a computational approach to obtain phonon heat capacity and heat transport rates between solids in various confined geometries that can be easily integrated into, e.g., the standard two-temperature model. Confinement significantly reduces heat capacity and may slow down heat transport. We validate our model with experiments on strongly disordered NbTiN superconducting nanostructure, widely used in highly efficient single-photon detectors, and we argue that confinement is due to their polycrystalline granular structure. These findings point to potential advances in cryogenic device performance through tailored material and microstructure engineering.