Understanding the units of wavelet time & frequency resolution
Confusion seems rooted in notation:

In $s = f_0 / (f \cdot \text{sampling rate})$, $f_0$ is the center frequency of mother $\psi$, and $f$ is the center frequency of the scaled $\psi$, i.e. $\psi_s$.
In $\sigma_{\hat\psi}=$ ..., $\omega_0$ is the (radian) center frequency of the wavelet of interest, or $\psi_s$, rather than $\omega_0=2\pi f_0$.
However, you can compute sigma of scaled from sigma of mother via: $\sigma_{\hat\psi_s} = \sigma_{\hat\psi} / s$ (see under 4.54).
$\text{sampling rate}$ is only relevant in discretized computations and must get canceled to yield physical units (Hz, sec, etc), so if you seek only physical use $s = f_0 / f_s$ (adjusted $f\rightarrow f_s$ to avoid confusion; recall that $f$ (or $\omega$) is what we integrate over, rather than plug in)
Do not involve $\text{sampling rate}$ in defining $\psi, \hat\psi$, scaled, or computing $\sigma$, or effectively most formulas you come across unless they explicitly include it, it's only to be used after computation, e.g. you first find $\sigma_{\hat\psi}$ without $\text{sampling rate}$, and then rescale as in $(4)$ here.
You commented &quot;$\sigma_\omega$ Hz&quot;, but it's actually &quot;rad-Hz&quot;; to get Hz do
$f=\omega / (2\pi)$.

Let me know if this makes sense (if not, again with a specific example).