You will need to reconstruct the sphere based on the uv coordinates on the mesh.
I would try the approach of finding spherically opposite uv coordinates within the mesh.
Best would be to find exact matches but I suspect you’ll need to interpolate the coordinates on a mesh face.
If you generate enough pairs that have good spread you can construct a centerpoint by analysing the lines connecting the pairs.
Another or additional approach would be to contruct planes through u.v. coordinates that should ly in the same plane.