It seems to me that the statement that Wu's experiment holds globally is equivalent to the statement that the universe has an orientation so I now tend to think the universe does not admit one of these orientation-reversing loops.
At first I was surprised that we could apparently draw a global conclusion based on local observations but I've mostly convinced myself now based on the following analogy (which I hope is not misleading).
Imagine we live in a two-dimensional universe and we want to know whether or not it is orientable. E.g., do we live on the surface of a torus or on the surface of a Klein bottle? We have defined a local orientation in the topologically-trivial patch where we live and so are used to speaking of "left-handed" and "right-handed" gloves. The question of whether these words are globally meaningful is exactly the question of whether our universe is orientable.
One day we learn about "colour" and are stunned to discover that all left-handed gloves are red and all right-handed gloves are blue. It seems very unlikely to us that red somehow transitions into blue as we traverse a loop and so we conclude we must live in an orientable universe.