Why does the Heisenberg uncertainty principle work in an atom?
Electrons don't have a well-defined orbital radius even in a hydrogen atom

An electron "orbiting" in a hydrogen atom does not have a well-defined position or even just a well-defined radius just because it has a well-defined energy.

The orbital is more like a 3D probability density function covering a cloudy space of a defined shape (spherical in the case of the lowest energy orbital but not of a single radius and extending with a finite probability from the nucleus to a vaguely defined boundary we often regard as the radius of the atom).

Different orbitals have different, increasingly complex, shapes defined by solutions of the Schrodinger equation. Heisenberg's uncertainly relationship limits our ability to define the location and momentum of the electron at the same time. The fuzzy probability clouds defined by Schrodinger's solutions are the best we can do.