I don't know whether this "at-at theory of motion" of Russell is related to the rest of your question.
Bohmian mechanics vs. orthodox QM
Firstly, it is important to note that Bohmian mechanics is a physical theory which is empirically equivalent to the orthodox recipe for making predictions. I.e., Bohmian mechanics gives the same predictions for all experiments one could come up with. Hence, if orthodox QM makes some predictions where the uncertainty principle plays a role, then Bohmian mechanics makes the same predictions. E.g., if one makes a sharp position measurement of an electron and then measures the (asymptotic) momentum, QM predicts that the statistics of the outcome of the momentum measurement will have at least a certain standard deviation if I do the experiment many times. Bohmian mechanics predicts the same statistics, so in this sense, the uncertainty relation does follow from Bohmian mechanics.
On the notion of hidden variables.
In your question, you said that Bohmian mechanics posits hidden variables. One has to be careful with the notion of hidden variables because there are several distinct meanings. The two most important are:
A theory that has ingredients (like particles, strings, fields) in addition to the wave function. These additional objects are sometimes called "hidden variables".
An assignment of actual (hidden) values to the observables/operators $\hat O \mapsto o$ of QM, such that a measurement of the observable $O$ will reveal the value $o$.
Now, Bohmian mechanics is a hidden variables theory in the first sense since there are particle positions $Q_i(t)$ in addition to the wave function $\Psi$. However, it is not a hidden variable theory in the second sense. Hidden variable theories of the second kind are not possible due to theorems like Kochen-Specker.
The term "hidden variables" is also misleading because the position of particles in Bohmian mechanics is not hidden at all. In fact, the positions are the only thing that can be observed in Bohmian mechanics. It is rather the wave function, which is kind of hidden.
On the notion of measurement in Bohmian mechanics.
It is also important that the notion of "measurement" does not play any role in the formulation of Bohmian mechanics. The better notion would be the "outcome of experiments". While orthodox quantum mechanics just builds the notion of measurement into its core, in Bohmian mechanics, one can analyze what precisely happens during an experiment called measurement of $O$. Observables or operators are just bookkeeping devices to handle the statistics of experiments. E.g., if one makes an experiment called measurement of $\hat O$, Bohmian mechanics predicts what the positions of particles of the detector needle will be. For a good account on this, see https://arxiv.org/abs/quant-ph/9601013.
At first glance, it might seem like a violation of the uncertainty principle, that particles have trajectories and therefore precise position $Q_i(t)$ and momentum $m_i \frac{\mathrm{d}}{\mathrm{d} t} Q_i(t)$. But this is not the case since the experiments we call measurement of $\hat P_i$ do not have $m_i \frac{d}{d t} Q_i(t)$ as output. In fact, one can show that there exists no experiment that reveals the instantaneous velocity $\frac{d}{d t} Q_i(t)$ of a Bohmian particle. See, for example, chapter 3. at https://link.springer.com/book/10.1007/978-3-031-09548-1 for details.
The reason is the following. In order to make an experiment that "measures the position" of a particle, one needs an apparatus that interacts with the particle. Consider the wave function of the apparaus_particle system $\Psi(Y,x)$ where $Y$ represents the degrees of freedom of the apparatus and $x$ the particle position. Interaction means that the time evolution does not factor into the $Y$ and $x$ subsystems (i.e., the Hamiltonian has an interaction part $H = H_Y + H_x + H_{Y,x}$. As a consequence of the uncertainty relation in the Schrödinger equation, any interaction $H_{Y,x}$ that leads to a "measurement of $x$", i.e, the Bohmian configuration $Q_Y(t)$ of the apparatus encodes the position of $x$, will drastically change the subsystem wavefunction $\Psi(Q_Y(t), x)$. Therefore, a second position measurement at a later time $t+\delta t$ will not yield the same result as if the first measurement was done at time $t+\delta t$. Therefore, you cannot have two consecutive precise experiments $x$ if the original Bohmian trajectory because the first experiment will destroy/alter the trajectory in such a way that the Bohmian velocity at time $t$, i.e $\dfrac{Q(t+\mathrm{d}t)-Q(t)}{\mathrm{d}t}$ does not match with the expression $\dfrac{Q(t+\delta t) -Q(t)}{\delta t}$ that we can measure. All of this works precisely in the way of the uncertainty relation.
Uncertainty relation vs. uncertainty principle
According to some views "uncertainty principle" is a fundamental feature that distinguishes quantum- from classical physics. (See, for example, https://arxiv.org/abs/1712.02894 for such a point of view). Instead of classical variables $X$, we have some fuzzy quantum variables $\hat X$ that have been quantized to satisfy the Heisenberg uncertainty relation. The principle would then state that everything "quantum" has this fuzziness.
In Bohmian mechanics, there is no "quantum fuzziness" in the formulation of the theory. The uncertainty is not built into the core of the theory but is a consequence of the dynamics of the theory and its statistical analysis. Once again, the quantum observables are only bookkeeping devices in Bohmian mechanics to efficiently describe the outcome of experiments.
I hope this answer might be helpful.
I think the articles https://arxiv.org/abs/quant-ph/9601013 and https://arxiv.org/abs/1712.02894 illustrate different views on Quantum theory, as they take opposite points of view.
