There's two layers of "nonlocality" at play here.
On the one hand, there's the issue of "action at a distance". Relativity tells us that you cannot have something affect something else faster than the speed of light. Note that here "affect" is to be understood as something along the lines of "there is no way to measure a system B and find that its state has been altered due to something that happened at a system A outside of its light cone".
On the other hand, quantum mechanics can display "nonlocal effects" of a different nature. Namely, there can be correlations between distant observers that cannot be explained via any kind of "shared classical correlation" (what we refer to as hidden variable theories in this context). They are "nonlocal" because measurements on a system can instantaneously affect entangled systems regardless of their spatial (or temporal, for that matter) distance. However, crucially, these nonlocal correlations are not at the level of measurement outcomes. There is no way to measure a system $B$ and be able to infer anything at all about what happened to another system $A$ outside of its light cone. There is no way to observe any effect on $B$ "caused" by anything outside of its light cone. This is also sometimes referred to as the no-communication theorem. See also Does Bell's theorem imply a causal connection between the measurement outcomes?.
Let me try and be even more precise about this: these nonlocal correlations I'm referring to are observable (otherwise they wouldn't be a significant part of the theory at all). But you cannot observe them performing any individual experiment. Meaning regardless of the shared state $A$ and $B$ have, whatever measurement they're doing, they won't be able to infer (superluminally) anything about the way the other party interacted with their state. However, if $A$ and $B$ repeat their experiment multiple times, changing the way they interact with their system each time, and then at the end go and compare all the measurement statistics they found, then they'll observe that the observed correlations are "nonlocal", meaning that the state they shared was "more correlated than classically possible".
Note that the above makes no reference to whether the quantum framework we're using is relativistic or not. In fact, the standard nonrelativistic formalism for quantum mechanics does allow superluminal communication, simply because special relativity is not taken into account at all in the formalism. But this is completely different from the issue of the nonlocal correlations due to quantum entanglement/correlations.
You can fix the relativistic issues by integrating relativity into the formalism, as you do in QFT, and then you get back the expected result that quantum mechanics is local (again, in the relativistic sense; the features about "nonlocal unobservable correlations" described in the first paragraph remain).
