According to the De-Broglie Wave Hypothesis an electron can be considered to be a wave. Red Shift occurs due to the expansion of space(Cosmological red shift), which must expand even on the scale of atoms, thus, even electron waves must be red shifted. However, that would imply that the wavelength of the electron wave is constantly increasing, and consequently its energy decreasing? But would that not cause the electron to spiral into the nucleus?
Not at all. Space is expanding, as in space is constantly being added. Although space might be added between the electron and nucleus, that does not effect the atom to significant degree. Its like having an atom in flatland and turning the flatland into a ball. the atom is not going to know much of a difference. If the sphere is returned to flat land the atom will then restore itself.
If the electron is in free motion away from a nucleus that is quite a difference. Then you might have aspects of what you mentioned.