What is the Compton wavelength of the electrons? Does that answer your question?
You may like to check out the link below, which addresses something of your question.
https://en.wikipedia.org/wiki/Electron_degeneracy_pressure
The uncertainty principle is an observational principle. Since 0K is unattainable in nature, the question of the applicability of the uncertainty principle at 0K does not arise.
Respected Remi Cornwall!
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The "Uncertainty principle (position & momentum" is not really a "principle", but a mathematical property of the Schrödinger equation (and possibly of other ones). It became a 'principle' after physics had moved from Germany to the US (Grass), together with the Copenhagen interpretation of QM. Landau was not part of that displacement, but - despite of his genius of its own (here, relativistic uncertainty relations, with Peierls, 1930) - was told to have sit at Bohr' feets.
Hence, look for the equations applying to your problem and the corresponding properties of them.
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