A search for Kaluza-Klein excited vector boson resonances, ${W}_{\mathrm{KK}}$, decaying in cascade to three $W$ bosons via a scalar radion $R$, ${W}_{\mathrm{KK}}\ensuremath{\rightarrow}WR\ensuremath{\rightarrow}WWW$, in a final state containing two or three massive jets is presented. The search is performed with $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ proton-proton collision data collected by the CMS experiment at the CERN LHC during 2016--2018, corresponding to an integrated luminosity of $138\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. Two final states are simultaneously probed, one where the two $W$ bosons produced by the $R$ decay are reconstructed as separate, large-radius, massive jets, and one where they are merged into a single large-radius jet. The observed data are in agreement with the standard model expectations. Limits are set on the product of the ${W}_{\mathrm{KK}}$ resonance cross section and branching fraction to three $W$ bosons in an extended warped extra-dimensional model and are the first of their kind at the LHC.