Recently, several groups have reported observations of collective modes of the charge order present in underdoped cuprates. Motivated by these experiments, we study theoretically the oscillations of the order parameters, both in the case of pure charge order and for charge order coexisting with superconductivity. Using a hot-spot approximation we find in the coexistence regime two Higgs modes arising from hybridization of the amplitude oscillations of the different order parameters. One of them has a minimum frequency that is within the single-particle energy gap and which is a nonmonotonic function of temperature. The other---high-frequency---mode is smoothly connected to the Higgs mode in the single-order-parameter region, but quickly becomes overdamped in the case of coexistence. We explore an unusual low-energy damping channel for the collective modes, which relies on the band reconstruction caused by the coexistence of the two orders. For completeness, we also consider the damping of the collective modes originating from the nodal quasiparticles. At the end we discuss some experimental consequences of our results.