I agree about the continuum only allowing an approximated simulation by Turing computation. But that’s orthogonal to QM, you already get that with classical fields. In Everettian QM, the evolution of the quantum state of the universe is still deterministic, following the Schrödinger equation. There is no probabilistic aspect in simulating it as a whole.

Regarding the continuum, the final word hasn’t been spoken with regard to the physical world, and probably also with regard to mathematics, since there is some level of controversy about the continuum even in the foundations of mathematics. We know for example that the continuum hypothesis is independent of ZFC, so what’s up with that. It might very well turn out that there is no continuum in physical reality. However, it seems highly unlikely that it’ll be anything like the naive conception of a cellular automaton.