One interpretation of setting $c=1$ is that we are measuring distances and times in "the same units." For instance, one way to set $c=1$ is to measure durations in years and distances in light-years. To undo such a choice, you multiply by an appropriate power of $c$.

The effect of $\hbar = c = 1$ is to use "the same units" for length, time, mass, and energy. It's pretty common (example, example) to hear people talk about "the dimension" or "the mass dimension" of an expression, which is basically how many factors of energy there are in its $\hbar=c=1$ units. For example, in order for the action $S=\int \mathrm d^4x\,\mathcal L$ to be dimensionless, the Lagrangian density $\mathcal L$ much have dimension four.

I find myself most frequently wanting to compare energies and lengths, for which it's useful to have $\hbar c = 0.197\rm\,GeV\,fm \approx \frac 15\,GeV\,fm$ in the back of my head. For example, I think a lot about the effective range parameter $r_0 = \hbar c / mc^2$ of a Yukawa potential for a force whose mediating particle has mass $m$. This parameter has dimension $-1$ and is the same as the Compton wavelength of the massive particle. There are fewer characters if I type $r_0 = \hbar / mc$, which is algebraically the same; however the cognitive load for me is lighter if I see $mc^2$ and think "energy" and then see $\hbar c$ and think "energy converts to length." The system is not overdetermined, so there is only one nontrivially correct way to convert using powers of $\hbar$ and $c$.

Your propagator $\frac1{p^2-m^2}$ has dimension $-2$, and your decay width $\Gamma = 1/\tau$ has dimension 1. A cross-section (dimensionally an area) must have mass dimension $-2$. You get the idea.

For coupling constants, it's nice if they are dimensionless, but that's not always how it works out. An example.

One interpretation of setting $c=1$ is that we are measuring distances and times in "the same units." For instance, one way to set $c=1$ is to measure durations in years and distances in light-years. To undo such a choice, you multiply by an appropriate power of $c$.

The effect of $\hbar = c = 1$ is to use "the same units" for length, time, mass, and energy. It's pretty common (example, example) to hear people talk about "the dimension" or "the mass dimension" of an expression, which is basically how many factors of energy there are in its $\hbar=c=1$ units. For example, in order for the action $S=\int \mathrm d^4x\,\mathcal L$ to be dimensionless, the Lagrangian density $\mathcal L$ much have dimension four.

I find myself most frequently wanting to compare energies and lengths, for which it's useful to have $\hbar c = 0.197\rm\,GeV\,fm \approx \frac 15\,GeV\,fm$ in the back of my head. For example, I think a lot about the effective range parameter $r_0 = \hbar c / mc^2$ of a Yukawa potential for a force whose mediating particle has mass $m$. This parameter has dimension $-1$ and is the same as the Compton wavelength of the massive particle. There are fewer characters if I type $r_0 = \hbar / mc$, which is algebraically the same; however the cognitive load for me is lighter if I see $mc^2$ and think "energy" and then see $\hbar c$ and think "energy converts to length." The system is not overdetermined, so there is only one nontrivially correct way to convert using powers of $\hbar$ and $c$.

Your propagator $\frac1{p^2-m^2}$ has dimension $-2$, and your decay width $\Gamma = 1/\tau$ has dimension 1. A cross-section (dimensionally an area) must have mass dimension $-2$. You get the idea.

For coupling constants, it's nice if they are dimensionless, but that's not always how it works out. An example.

You ask in a comment about the branching ratio

$$\frac{G_F^2m^9\tau}{m^4\hbar} $$

which should be dimensionless, like any branching ratio. I would process this by turning every sub-expression into energies and lengths:

$$ \left(\frac{G_F}{(\hbar c)^3}\right)^2 \frac{(mc^2)^9}{(mc^2)^5} \frac{c\tau}{\hbar c} $$

The Fermi constant is usually tabulated with the factors of $\hbar c$ which put it into energy units, where it has dimension $-2$. For most particles the rest energy $mc^2$ is easier to locate than the mass $m$ in SI units. And I find it less confusing to identify ${c\tau}/{\hbar c}$ as an expression with energy dimension $-1$ than I do the equivalent expression $\tau/\hbar$ --- perhaps because $\hbar c$ has a reasonable value in energy-length units.