There is no contradiction with the behavior of nonlinear classic systems described by ODEs.
The crucial point in Wald’s discussion is the use of a compact region.
The idea in quantum field theory is the following one. One is solving a (generally non linear but normallyhyperbolic) PDE for a field say $\phi=\phi(x)$, assigning initial data (possiby with compact support) on a Cauchy surface $S$, and considers the unique solution arising from the existence theorem. (Without an existence theorem, at least local, what follows is meanigless.)
Next one focuses on that solution restricted to a compact region $K$ of the spacetime, included in the domain of definition of the considered solution.
What is expected from physics, making more explicit Wald’s comments, is that slightly varying the initial conditions:
(a) the new solution $\phi'$ is still defined on a region which includes $K$ if its initial condition is sufficiently close to the one of $\phi$ (in a suitable norm as in (b) below);
(b) we can have $||\phi-\phi'|| <\epsilon$ in $K$ for some meaningful norm defined in $K$ and for any arbitrarily small $\epsilon>0$, if the respective initial conditins satisfy $||\phi_0-\phi'_0||_S< \delta$ for a sufficiently small $\delta>0$.
Failure of (a) concerns existence of runaway solutions: the field diverges before reaching the boundary of some $K$ also for small changes in the intial conditions.
Failure of (b) means that, though both the solutions exist in $K$, we do not have a good control of them by varying the initial conditions.
Let us move on to consider standard ODEs instead. Here (a) and (b) cannot take place as I am going to discuss, and thus Wald’s requisites are automatically fulfilled though it does not seem at first glance in view of known behavior of chaotic systems. Actually this behavior is not in contradiction with Wald’s requirements as I discuss eventually.
Now the relevant notion of domain is along the temporal axis, where the parameter $t$ of the solutions ranges. The relevant compact is a set of the form $[T_1,T_2]\ni t$.
Assuming that the ODE is in normal form, we can always transform it into a 1st order (system of) ODE(s) by adding a sufficienly large number of trivial further variables. At this juncture one can exploit the general theory of local one-parameter groups (local flow) on manifolds, here a suitable jet bundle $M$:
$$\Phi= \Phi_t(p)\:,\quad (t,p) \in A \subset \mathbb{R}\times M$$
where
$$\Phi_t(p)= x(t|p)$$
is the (unique) maximal solution of the system with initial condition $x(0)=p$.
Under suitable (and natural) hypotheses on regularity of the right hand side of the ODE in normal form, we see that
the pathology in (a) is not possible because the joint domain $A$ of the local group is always open and we can use obvious properties of the product topology;
the pathology in (b) is not possible as well because the local group
$$A\ni (t,p)\mapsto \Phi_t(p)\in M$$
is always (at least) jointly continuous (and thus uniformly continuous on compacts).
What may happen, apparently in contradiction with what we said above (but which is actually a physically admissible behavior of non-linar systems) is the following.
Supposing that the domain of every solution is the whole real axis (i.e. the flow is complete and $A=\mathbb{R}\times M$), the condition $$||x(t)-x'(t)||<\epsilon$$ cannot be fulfilled uniformly in $t\in (-\infty,+\infty)$, independently from the value $\delta>0$ chosen in $$||x(0)-x'(0)||< \delta.$$
Even if the initial conditions are arbitrarily close to each other, if waiting a sufficiently large lapse of time the solutions may considerably diverge from each other.
The crucial point is that here we are dealing with a non compact interval, given by the whole linear line.
This is, for instance part of what happens of the Lorenz system and denotes the so called "sensitive dependence on initial conditions".
Note that this is not sufficient for declaring that the system exhibits chaos, just necessary.
