I have heard recently that the de Broglie-Bohm theory, or pilot wave theory, is an acceptable alternative to the Copenhagen interpretation. But how does it explain Heisenberg's uncertainty principle? Doesn't uncertainty depend on the Copenhagen interpretation?
It's exactly the same in Bohmian mechanics, only the reasoning is different.
From Dürr et al. (1992) - DOI: 10.1007/BF01049004:
From a general perspective, perhaps the most noteworthy consequence of our analysis concerns absolute uncertainty (Section 11). In a universe governed by Bohmian mechanics there are sharp, precise, and irreducible limitations on the possibility of obtaining knowledge, limitations which can in no way be diminished through technological progress leading to better means of measurement. This absolute uncertainty is in precise agreement with Heisenberg's uncertainty principle. But while Heisenberg used uncertainty to argue for the meaninglessness of particle trajectories, we find that, with Bohmian mechanics, absolute uncertainty arises as a necessity, emerging as a remarkably clean and simple consequence of the existence of trajectories. Thus, quantum uncertainty, regarded as an experimental fact, is explained by Bohmian mechanics, rather than explained away as it is in orthodox quantum theory.
Essentially, as Bohmian mechanics is deterministic and relies on a universal wave function, it's impossible to separate our measuring equipment (or ourselves) from the quantity being measured, hence it's never fully in equilibrium, which gives rise to the uncertainty.