Does local physics depend on global topology? Motivating Example
In standard treatments of AdS/CFT (MAGOO for example), one defines $\mathrm{AdS}_{p+2}$ as a particular embedded submanifold of $\mathbb R^{2,p+1}$ which gives it topology $S^1\times \mathbb R^{p+1}$ where the $S^1$ factor is "timelike."  This leads to the property that there are closed timelike curves, so to obtain a causal spacetime, one "decompactifies" the timelike direction by unwrapping the $S^1$ factor to $\mathbb R^1$ while retaining the same expression for the metric (in other words one considers the universal cover of $\mathrm{AdS}_{p+2}$
It seems to me that since the metric has not changed, but the topology has, that at least at the classical level, one cannot locally distinguish between $\mathrm{AdS}_{p+2}$ and its universal cover.
The Question
Can one locally distinguish between the two at the quantum level?
Namely, if we were working with fully quantum AdS/CFT, would quantum effects due to global topology become apparent in local physics?  I would think that the answer is yes in some sense based on intuition from a free particle moving in 1D on a circle, for example, versus on the entire real line.
Apologies for the vagueness of parts of the question.
 A: OK, let us start from your example. I think that it is too pathological to be considered as a safe starting point for this discussion, which is worth and interesting however. Nevertheless I would like to spend some words about this case since it permits to introduce some general issue useful in the second part of my answer.
AdS_n is not globally hyperbolic. Globally hyperbolic, more or less, means that there are spacelike smooth surfaces, called Cauchy surfaces,  where the assignment of initial data assures both  existence and uniqueness of the solution of any field equation of hyperbolic type (linear or almost linear) for fields propagating in the considered spacetime. This failure of  AdS_n is due to a pair of pathologies: (1) the existence of closed timelike curves and (2) the absence of Cauchy surfaces. The former is very strong, since it does not allow to build up  a well-behaved classical field theory. For instance, it is very difficult to define elementary tools as advanced, retarded and causal propagators. I am not saying that there are no chances to extend these general notions to this case, I am rather saying that the physical interpretation of the results is questionable. As you suggest, one may unwrap the time direction passing to the universal covering and getting rid of (1). The problem (2) remains however. Nevertheless it is possible to extend the standard formalism, valid for globally hyperbolic spacetimes, to this case too. As a matter of fact, it has been done in the past by several authors (in particular I remember a quite long and detailed paper by Bob Wald on these issues). But this way leads to just one spacetime and not a pair of spacetimes with different global topologies, so  your issue cannot be raised any more here.   
Abandoning AdS spacetimes, and sticking to the whole class of globally hyperbolic spacetimes, it make however sense to ask if the global topology can affect local quantum properties. If $M$ is such spacetime, its topology, as is known, is always of the form $R \times S$, where $S$ is a spacelike submanifold diffeomorphic to every Cauchy surface in $M$. (Notice that the metric does not splits as the topology does, so $M$ is diffeomorphic to $R\times S$ but, in general, not isometric to it.) In view of this particular product topology, your question concerns the topology of $S$: is it possible that its topology has some effect locally when dealing with quantum fields? For instance we can consider a couple of globally hyperbolic spacetimes whose respective topologies are $R\times S$ and $R \times \widetilde{S}$, where $\widetilde{S}$ denotes the universal covering of the multiply connected manifold $S$.
In my opinion the answer is YES. However we have to sharply distinguish between properties of quantum observables and properties of quantum states to answer. Indeed, the answer is positive regarding the second class of objects only. 
Quantum observables are, roughly speaking, quantum fields. I include in this category things like renormalised objects as Wick polynomial $\phi^n(x)$ (or similar, more complicated, local objects carrying properties as spin and charge). The UV renormalization procedure for a quantum field theory in curved spacetime (so the metric enters the field equations but remains a classical field) can be performed without referring to a reference state, as established in the last decade using the algebraic approach. It was done extending the so-called Epstein-Glaser approach to renormalization  from Minkowski spacetime to a general globally hyperbolic spacetime.  (Obviously the procedure reduces to the standard one in Minkowski spacetime.)
The observables obtained this way are generally locally covariant.        It means that if you are given a couple of spacetimes, $M$ and $M'$,
such that $M$ can be isometrically embedded in $M'$ preserving the causal structures and time orientations, also the algebra of quantum observables of $M$ turns out to be embedded in that of $M'$ through a suitable morphism of $^*$-algebras. In other words we cannot distinguish between the quantum observables of $M$ and $M'$ if looking at $M$, viewing the spacetime $M$ as a subregion of $M'$ or on its own right. It is worth stressing that all UV renormalisation ambiguities are  encompassed  in some terms depending on the local curvatures only, so even renormalisation of local observables cannot  help to positively answer your question.
It remains to focus on states. The allowable states (those permitting the perturbative  renormalization procedure) are the so called Hadamard states.
In practice they are defined by requiring that they are of Gaussian type (so that the two-point function determines the whole class of n-point functions) and that their short distance structure resembles that of Minkowski vacuum. (There is a much more technically useful characterization arising form  micro local analysis,  but it is irrelevant here.)
I stress  that the constraint on short distances fixes only the UV singularity in terms of the local geometry, while the remaining part  of the state is free and it is therefore sensible to the global topology of the spacetime.   There is a huge class of Hadamard states for a given spacetime. When one computes objects like $<\phi^2(x)>$, the UV divergences are removed by subtracting the universal UV divergence of Hadamard states which depends on the local geometry only. Consequently,  global effects may appear at this stage. For instance, if you consider Minkoski spacetime $M = R \times R^3$ and another spacetime $M'$ obtained by identifying the opposite faces of a 3-cube in $R^3$, you have a pair of globally hyperbolic spacetimes which locally are identical. Thus the UV divergence of Hadamard state coincide. $M$ and $M'$ admit a corresponding pair of preferred Gaussian Hadamard states for the real scalar Klein-Gordon field, singled out by the property to be invariant under the complete class of isometries of the respective spacetime, and the fact that they are ground states with respect to a Hamiltonian operator associated with a natural (isometric) notion of time evolution. These states are, respectively, the standard Minkowski vacuum $< >_M$ and a similar state in $M'$, $< >_{M'}$,obtained from the former exploiting for instance the method of images.   If you compute $<\phi^2(x)>_M$ and $<\phi^2(x)>_{M'}$  you obtain two different values even if you subtract the same divergence. The difference takes global topological properties into account. These differences are somehow related to the Casimir effect when you consider the stress energy tensor in place of $\phi^2$.
