it is true that the state is assigned to spacelike surface, and the "update" takes place instantaneously on that surface.
No. This is a noncovariant, observer-specific view.
In the covariant, observer-independent view of fields, states are labeled instead by the causal classical solutions of hyperbolic field equations. On the collection of these the Peierls bracket is defined, which is the covariant version of the Poisson bracket. Each observer picks out a particular frame and with it at each time a Cauchy surface that intersects a causal classical solution exactly once - giving the instantaneous field labels of the observer's state satisfying a functional Schroedinger equation that reduces after space discretization to the nonrelativistic treatments.
Thus the same covariant state appears different to each observer, just as in classical relativistic physics the same covariant length appears different to different observers.
In classical relativistic physics, one can directly compare only events modeled by a single observer; models of different observers have no connection unless they agree on the information encoded in it and use the rules of relativity to translate their models into mathematically equivalent things. The same holds even more so in quantum relativistic physics.
The collapse is a sudden change of the model used by an observer to reinterpret the situation when new information comes in, hence depends on when and whether the observer cares to take notice of a physical fact. This clearly cannot affect other observers and their models of the physical situation. Hence there is no nonlocal effect. Nonlocal correlations appear only when a single observer compares records of other (distant) observer's measurements. At that time the past light cone of this observer contains all the previously nonlocal information, so that locality is again not violated