I am looking for a way to translate formulas written in natural units into either HLU units or SI units. Seeing the Planck constant and the speed of light would help me understand what is going on. The problem is that the advice to use dimensional analysis suggested fiddling with multiplying or dividing until it works. It was not a clearly defined procedure. Where can I find clear instructions how to rewrite a formula in natural units so that the constants will become visible?
What you are looking for is called dimensional analysis.
You multiply every but one term of the equation with unknown powers of $c^\alpha$ and $\hbar^\beta$, where $\alpha$ and $\beta$ are variables. Then you identify the powers of mass, length, time etc. on both sides, which should give you equations in $\alpha$ and $\beta$. After solving them, you know exactly how many $c$ and how many $\hbar$s you have to add.
This works because we are not changing the equation. In natural units, $c=1$ and $\hbar=1$ (and maybe some other constants as well) and therefore multiplying by $1^\alpha$ does not change anything.