= Solution
Work in <natural units> $\hbar=c=k_B=1$, at <temperatures> far below the $W$-boson mass but of order or above the <Electron> mass. The <Fermi interaction> then supplies a local four-fermion amplitude. The <Fermi constant> has dimension energy$^{-2}$. In scattering of a thermal lepton from a nonrelativistic nucleon, the nucleon current matrix element has size $m_N$, while the lepton current has size $T$. Thus $\mathcal M\sim G_Fm_NT$. The two-body <relativistic scattering cross-section> contains the kinematic factor $s^{-1}\sim m_N^{-2}$; when the final and initial lepton energies are comparable, their momentum ratio is of order one. Consequently $\sigma\sim|\mathcal M|^2/m_N^2\sim G_F^2T^2$. This estimates <temperatures> above the <Electron> mass and mass-splitting thresholds; those scales only affect the dimensionless rate function near freeze-out.
The light leptons in the thermal bath have <number density> $n\sim T^3$ and speed of order unity. Their conversion rate per nucleon is therefore
$$
\boxed{\Gamma_{n\leftrightarrow p}\sim n\langle\sigma v\rangle\sim G_F^2T^5.}
$$
For example, $n+\nu_e\leftrightarrow p+e^-$ and $n+e^+\leftrightarrow p+\bar\nu_e$ change the <neutron-to-proton ratio>. The bath density here is the lepton density, not the much smaller <baryon> density. Dimensionless weak couplings, phase-space integrals and the several reaction channels set a numerical coefficient. <Electron> masses and the neutron-proton splitting also become important for a precision rate near <cosmological weak freeze-out>.
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