There are two significantly different notions of nonlocality in use - the violation of causal locality giving rise to equal-time causal commutation rules and the violation of Bell inequalities and the like. Let me call the former causal nonlocality and the latter Bell nonlocality.
More precisely, causal locality is a condition ensuring that signals, matter, and energy cannot travel faster than light. On the other hand, Bell locality is the assumption that the state of an extended system factors into the states of localized parts of the system. Roughly speaking, this means that complete information about the state of region A and complete information about the state of region B is equivalent to complete information about the union of regions A and B, and this information propagates independently if A and B are disjoint.
The condition characterizing Bell locality is satisfied for classical point particles but not for classical coherent waves extending over the union of A and B. The Maxwell equations in vacuum provide examples of the latter, although they satisfy causal locality. Thus causal locality and Bell locality are two essentially different concepts.
According to our present knowledge, causal nonlocality is not realized in the universe. If it were, it would wreck the basis of all our subatomic quantum field theory. I nowhere claim (or see a claim of) anything that could support causal nonlocality.
On the other hand, Bell nonlocality has been amply demonstrated experimentally for quantum particles (photons, electrons, and even small molecules). It is an intrinsic effect of approximating the analysis experiments whose fundamental description would need quantum field theory by a simpler analysis in terms of a few particle picture. Bell locality applies to particles only and loses its meaning for fields, which are intrinsically Bell nonlocal, even classically.
Thus quantum physics is local in the causal sense but nonlocal in the Bell sense.