Has this gone through any peer review process?
No. Stephen Wolfram is wealthy. He doesn't need to operate or publish within the academic system.
I glance at his writings over the past few years and most often think that unless he uses his theory to solve a problem others care about, his line of research will be abandoned after he passes.
I did buy and read most of it, as I also have bought many of Penrose's popular books, so perhaps I'm just a disgruntled purchaser.
As for how non-local entanglement can occur, I'll just offer my current speculation, having spent about a year working on this kind of approach. You'll have to give up SR in its traditional sense. There will be a preferred reference frame, which is the rule application order that the automata uses in any particular history. The challenge becomes to explain why it is not observable and you have approximate Lorentz covariance. I think this will be less hard than one imagines -- even in Feynman's lectures you see explanations of SR that involve an aether (or a global frame if you prefer) and clocks defined by light bouncing between mirrors moving in this aether. Of course all physical laws have to be covariant, and explaining how this happens requires you to know how the physical laws are "microphysically". But the graph itself in graph-automata models is a good candidate for an ether, with particles being e.g. topological defects. Somehow covariance must reflect how a dynamical account of defect behavior changes as one foliates rule application order, mysterious but not inconceivable.
Now, entanglement: one imagines entanglement is implemented by long-range connections, which in graph-type models could take many forms. This is a kind of discrete version of the "ER = EPR" proposal. But they will have to be such a limited form of connection that they do not permit signaling, and I think they way they can do this is via some sort of knot-theoretic braiding. Only be comparing measurement outcomes classically will it be possible to deduce the way the braiding was effected by measurement and confirm you had e.g. a GHZ state.
Now, QM is more than just entanglement, but in the words of Jaynes: "QM is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature - all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble." When scrambled, all the ingredients look inextricably connected. I think the unscrambling will seem beyond hope until one has the exact recipe to recreate the omelette.
That doesn't mean what he's saying has any value. I'm not saying what he's written here doesn't, but the prodigiously smart are just as capable of being intellectually lazy as anyone else. Anyone making grandiose claims like "our Physics Project [... is] showing us something even bigger and deeper: a whole fundamentally new paradigm for making models and in general for doing theoretical science" doesn't get to rest on their laurels if they want to be taken seriously.