Since the electron is in a fuzzed-out orbit because of quantum effects, it'll involve Planck's constant hbar. Since the proton is 1836 times as heavy as the electron, one can be approximate it quite well by being fixed, so the Bohr radius WON'T depend much on the proton mass. This is not a relativistic calculation so we won't use c. To combine these to get a length, we fiddle around. Half an angstrom, in other words.
Angstroms are the length scale of atomic physics.
Now in the hydrogen atom Delta x is just the Bohr radius and we can write Delta p, the uncertainty in momentum, as m Delta v, where v is velocity. But one can make it utterly precise, as Schrodinger did. A simple way to think of it is this. It's easy to guess how big the Compton wavelength is using the knowledge that it depends only on the mass of the electron, relativity and quantum mechanics. These two constants enable us to express units of mass in terms of dimensions of inverse length.