Coherent properties of vortex conical waves propagating through a turbulent atmosphere are theoretically studied with the use of the analytical solution of an equation that describes the evolution of the second-order transverse mutual coherence function of an optical radiation field. The following parameters of vortex conical waves are considered: the degree of coherence, the coherence radius, the integral scale of the degree of coherence, and the integral scale of the squared degree of coherence. The effect of atmospheric turbulence on these coherence characteristics of vortex conical waves is analyzed at different values of their parameters. It turns out that the degree of coherence of a vortex conical wave, formed from a Gaussian beam while passing through a conical lens (axicon) and a spiral phase plate, at its optical axis, is almost independent of the initial radius of the Gaussian beam and the radius of the axicon aperture. In addition, all the coherence characteristics of vortex conical waves depend on the topological charge stronger than on the wave-vector component normal to the radiation propagation direction. A meter of the integral scale of the degree of coherence of vortex Bessel-like optical beams is shown to be a preferred sensor of optical radiation distortions in a turbulent atmosphere as compared to a meter of the coherence radius of such beams. A lower degree of coherence of vortex conical waves than of fundamental (vortex-free) conical waves in a turbulent atmosphere is proven with the use of the integral scale of the degree of coherence of these optical waves as a referent criterion.