The electron spectral shapes corresponding to the low-Q ${\beta }^{-}$-decay transitions ${}^{151}\text{Sm}(5/2_{g.s.}^{-})\to {}^{151}\text{Eu}(5/2_{g.s.}^{+})$, ${}^{151}\text{Sm}(5/2_{g.s.}^{-})\to {}^{151}\text{Eu}(7/2_1^{+})$, ${}^{171}\text{Tm}(1/2_{g.s.}^{+})\to {}^{171}\text{Yb}(1/2_{g.s.}^{-})$, ${}^{171}\text{Tm}(1/2_{g.s.}^{+})\to {}^{171}\text{Yb}(3/2_1^{-})$, ${}^{210}\text{Pb}\left(0_{g.s.}^{+}\right)\to {}^{210}\text{Bi}\left(1_{g.s.}^{-}\right)$, and ${}^{210}\text{Pb}\left(0_{g.s.}^{+}\right)\to {}^{210}\text{Bi}\left(0_1^{-}\right)$ have been computed using beta-decay theory with several refinements for these first-forbidden nonunique (ff-nu) ${\beta }^{-}$ transitions. These ff-nu ${\beta }^{-}$ transitions have non-trivial electron spectral shapes with transition nuclear matrix elements (NMEs) computed by using the microscopic Interacting Boson-Fermion Model (IBFM-2) for the decays of $^{151}$Sm and $^{171}$Tm, and the nuclear shell model (NSM) for the decay of $^{210}$Pb. Within the respective Q windows, the computed ff-nu electron spectral shapes deviate maximally at sub-percent level from the universal allowed shape, except for the transition ${}^{210}\text{Pb}\left(0_{g.s.}^{+}\right)\to {}^{210}\text{Bi}\left(1_{g.s.}^{-}\right)$, where the maximal deviation is some 2.7%. This confirms that the so-called ξ approximation is fairly good for most of these low-Q ${\beta }^{-}$ transitions and thus the allowed shape is a rather good first approximation. Our computed spectral shapes could be of interest for experiments aiming to measure the cosmic neutrino background (CνB), like the PTOLEMY experiment. We have also derived CνB cross sections for the ground-state transitions of the considered nuclei at the β endpoint. Our findings indicate that more work on the atomic mismatch correction is needed in the future in order to extract reliable and precise CνB cross sections for any nuclear target.