What is the meaning of uncertainty principle in the case of hidden-variable theories? What would be the meaning of the uncertainty principle in the case of hidden-variable interpretations of quantum mechanics, especially contextual hidden-variable theories?

Edit: Another way to put the question would be whether it is possible to circumvent the uncertainty principle within the framework of contextual QM.
If not, why is there still irreducible uncertainty in a deterministic theory?
Doesn't this mean that the observational part of the wave function (Quantum Information-al!) is responsible for the uncertainty?
Even more, do we need to inevitably resort to the Observer Effect interpretation (aka noise-disturbance) of the Uncertainty Principle to explain such an irreducible uncertainty within the framework of a deterministic theory?
 A: I think you are asking what is the hidden-variable interpretation/description of unsharply defined observables when, in the standard interpretation of QM, these observables are incompatible and the quantum state is not eigenstate of both variables.
For systems quantistically defined in finite dimensional Hilbert spaces, existence of non commuting elementary observables  already implies contextuality  in realistic hidden variable interpretations of QM, in view of the Kochen Specker theorem.
That theorem assumes  hypotheses on the (non-quantum) valuation functions
$v_\lambda(A)$ only for pairs of compatible observables  $A,B$ and proves that the theory must be contextual. That is the powerfulness of the theorem.
How the quantum phenomenology of uncompatible observables is described in hidden variable theories is matter of each concrete hidden variable theory (independently from the fact that all these theories must be contextual).
Contextuality alone  is by no means able to provide an intepretation of the quantum phenomenology of incompatibke variables, it is only a necessary constraint.
Presumably, a description  should be related to the epistemic (vs ontic) stochasticity of the hidden-variable state in a hidden variable theory. In other words an explanation should be given in terms of the probability distribution which describes our knowledge of the hidden variable $\lambda$.
A problem is the the (standard version of the) KS theorem applies to finite dimensional Hilbert spaces and bounded observables. Therefore it cannot be directly applied to, e.g.,  a particle on the real line (where the Heisenberg principle applies!)  without stronger hypoteses on the set of observables and on the valuation function.
However, you can focus on hidden variabe theories explicitely constructed to deal with this case. I know only one of them: the Bohm theory. It is contextual and  it includes an explanation of the Hesinberg principle
A: The uncertainty principle has the exact same meaning with hidden-variable interpretations as with any other interpretation of QM. It says that in order to reproduce the predictions of QM, the probability distribution over the hidden variables must be such that the product of the statistical uncertainties of the measured value of certain observables satisfy certain lower bounds.
There might be some philosophical sense in which the quantum state isn’t “truly” uncertain for hidden-variable theories, but the uncertainty of what result will be measured simply reflects our classical lack of knowledge of the exact state of the hidden variables. But the practical relevance is the same - the mathematical mapping between the state of the hidden variables and the outcome of measuring observables must be such that certain inequities are satisfied.
A: The Hidden Variable Interpretation does not rule out probabilities. It just states that QM is not truly random and there is a parameter we do not know about. And since right now we have found no hidden parameter, we still don't know where the particle would be. That means probability density is still probability density, and uncertainties are still uncertainties. It's just that in this interpretation, probabilities and uncertainties are subjective.
