Unsteady natural convective flows of nanofluids in a vertical channel with circular cross-section have been investigated using the integral method transforms. The wall temperature on the cylindrical surface is prescribed as a given time-dependent function. The thermal transport is considered of Cattaneo type. Analytical solutions for the fluid temperature and velocity are determined by using the Laplace transform with respect to the time variable and finite Fourier transform with respect to the azimuthal coordinate. The inverse Laplace transforms are obtained using the residues theorem in complex analysis. Solutions in the general case are suitable for particularizations for several azimuthal angles of the channel. As a practical case of these flows, the flow of the alumina-water nanofluid is investigated when the cylindrical surface of the channel is kept at constant temperature. The influence of nanoparticles volume fraction on fluid temperature and velocity is studied. Numerical simulations give optimal values of the nanoparticle volume fraction for the enhancement of the thermal transport. This study would be useful in choosing the optimal value of the nanoparticles volume fraction to improve the thermal transport.