We examine the Weyl double copy relation for vacuum solutions of the Einstein equations with a cosmological constant using the approach we previously described, in which the spin-1/2 massless free-field spinors (Dirac-Weyl fields) are regarded as basic units. Based on the exact non-twisting vacuum type N and vacuum type D solutions, the finding explicitly shows that the single and zeroth copies fulfill conformally invariant field equations in conformally flat spacetime. In addition, irrespective of the presence of a cosmological constant, we demonstrate that the zeroth copy connects Dirac-Weyl fields with the degenerate electromagnetic fields in the curved spacetime in addition to connecting gravity fields with the single copy in conformally flat spacetime. Moreover, the study also demonstrates the critical significance the zeroth copy plays in time-dependent radiation solutions. In particular, for Robinson-Trautman (A) gravitational waves, unlike the single copy, we find that the zeroth copy carries additional information to specify whether the sources of associated gravitational waves are time-like, null, or space-like, at least in the weak field limit.