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The electron spectral shapes corresponding to the low-$Q$ $β^-$-decay transitions $^{151}$Sm$(5/2^-_{\rm g.s.})\to\,^{151}\textrm{Eu}(5/2^+_{\rm g.s.})$, $^{151}$Sm$(5/2^-_{\rm g.s.})\to\,^{151}\textrm{Eu}(7/2^+_{1})$, $^{171}$Tm$(1/2^+_{\rm g.s.})\to\,^{171}\textrm{Yb}(1/2^-_{\rm g.s.})$, $^{171}$Tm$(1/2^+_{\rm g.s.})\to\,^{171}\textrm{Yb}(3/2^-_{1})$, $^{210}\textrm{Pb}(0^+_{\rm g.s.})\to\,^{210}\textrm{Bi}(1^-_{\rm g.s.})$, and $^{210}\textrm{Pb}(0^+_{\rm g.s.})\to\,^{210}\textrm{Bi}(0^-_{1})$ have been computed using beta-decay theory with several refinements for these first-forbidden nonunique (ff-nu) $β^-$ transitions. These ff-nu $β^-$ 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 $^{210}\textrm{Pb}(0^+_{\rm g.s.})\to\,^{210}\textrm{Bi}(1^-_{\rm g.s.})$, where the maximal deviation is some 2.7$\%$. This confirms that the so-called $ξ$ approximation is fairly good for most of these low-$Q$ $β^-$ 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 (C$ν$B), like the PTOLEMY experiment. We have also derived C$ν$B 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 C$ν$B cross sections for any nuclear target.