We investigate the role of dense Mpc-scale environment in processing molecular gas in distant Low luminosity radio galaxies (LLRGs) in galaxy (proto-)clusters. We have selected within the COSMOS and DES surveys a sample of five LLRGs at $z=0.4-2.6$ that show evidence of ongoing star formation on the basis of their far-infrared emission. We have assembled and modeled the far-infrared-to-ultraviolet spectral energy distributions (SEDs) of the LLRGs. We have observed the sources with the IRAM-30m telescope to search for CO emission. We have then searched for dense Mpc-scale overdensities associated with the LLRGs using photometric redshifts of galaxies and the Poisson Probability Method, that we have upgraded using the wavelet-transform ($mathit{w}$PPM), to characterize the overdensity in the projected space. Color-color and color-magnitude plots have been derived for the fiducial cluster members. We set upper limits to the CO emission of the LLRGs, at $z=0.39, 0.61, 0.91, 0.97$, and $2.6$. For the most distant radio source, COSMOS-FRI 70 at $z=2.6$, a hint of CO(7$rightarrow$6) emission is found at 2.2$sigma$. The upper limits found for the molecular gas content $M({rm H}_2)/M_star<0.11$, 0.09, 1.8, 1.5, and 0.29, respectively, and depletion time $tau_{rm dep}lesssim(0.2-7)$ Gyr of the five LLRGs are overall consistent with the values of main sequence field galaxies. Our SED modeling implies large stellar masses for the LLRGs, in the range $log(M_star/M_odot)=10.9-11.5$, while the associated Mpc-scale overdensities show a complex morphology. The color-color and color-magnitude plots suggest that the LLRGs are consistent with being star forming and on the high-luminosity tail of the red sequence. The present study increases the limited statistics of distant cluster core galaxies with CO observations. The radio galaxies of this work are excellent targets for ALMA and JWST.
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