The theory only deals with the local curvatures, not the global topology. Hence any manifold with an allowed metric is allowed. These can be infinitely many, especially for negative curvature space-times.
So I suppose for the same physical situation, there would be many manifolds from which it can be studied, then, how do we choose on which manifold that we actually do the calculations?/ find the manifold that fits for that scenario?
As I understand your question, you are asking - "What local effects if any, are present in GR using which we can distinguish between manifolds (representing spacetime) that are same locally but not globally."
Let me state the definition of a manifold quickly - A topological manifold $M$ is a topological space where every point $p$ of $M$ lies in some open set $U$ such that that open set can be mapped to some region of $\mathbb{R}^{d}$. This map called $x$ should be invertible and both $x$ and $x^{-1}$ should be continuous. In other words if $M$ is locally homeomorphic to $\mathbb{R}^{d}$ then M is a topological manifold.
If $M$ represents spacetime, then we can only probe $M$ within this finite region $U$. We can then probe the topology of $U$ and, due to the homeomorphic nature discussed above, understand the local topology of $M$ itself. But we can never know the global topology of a generic manifold $M$.
In GR, further to locally homeomorphic property, we demand an even stronger condition - smoothness and locally diffeomorphic.
Now, if a collection of manifolds are locally diffeomorphic to $\mathbb{R}^d$ (and therefore to each other), then we (local observers) would not be able to distinguish one from the other. So, to answer your question - in my opinion, there are no effects that we can observe locally that will inform us about the global nature of $M$ (with a possible exception of globally hyperbolic spacetimes). Hence, all local effects would be identical.
