In a fully deterministic universe, every time-slice can be considered equivalent to a boundary condition, and the concept of a speed limit of propagation still makes sense. In the instant after the considered slice, a point is affected only by its immediate neighbours. But as we keep going further in time, to calculate the state of a point we will need to account for larger and larger neighbourhoods of the same point in the first slice.
This is exactly what happens e.g. in general relativity, which fully deterministic (away from singularities) yet speed-limited.