Low-dimensional materials are susceptible to electronic instabilities such as charge density waves (CDWs), originating from a divergence in the Lindhard electron response function, combined with a finite electron-phonon coupling strength. In this paper, we present a detailed characterization of the CDW in the quasi-one-dimensional material CuTe, including (1) direct visualization of lattice distortion seen with noncontact atomic force microscopy in real space, (2) the out-of-plane momentum dependency of the CDW gap size of the quasi-one-dimensional bands by angle-resolved photoemission spectroscopy, and (3) coherent dynamics of a photoexcited phonon mode seen by time- and angle-resolved photoemission spectroscopy, with frequency and wave vector qCDW corresponding to the soft phonon modes predicted by theory. Furthermore, we find that the CDW gap closes through a transient band renormalization. We thus confirm that, despite the quasi-one-dimensional characteristics of CuTe, it hosts inherently three-dimensional CDWs.