Here is a screenshot from nakahara. Now, to me it looks like that the uniformization theorem implies that all 2d manifolds are conformally flat, because the constant curvature metrics described in (14.119) are conformally flat. Is my conclusion correct ?
All of the constant 2d curvature metrics are locally conformally equivalent (so conformally flat if you wish), but the closed 2d manifolds are not all globally conformally flat. You can see this for instance from the Gauss-Bonnet Theorem, which tells us that the total curvature of a closed 2-manifold is (proportional to) the Euler characteristic. So in particular spheres and surfaces of genus $g \ge 2$ can never carry a global flat metric. There is a concept in low-dimensional topology of a Euclidean cone metric, which is a singular metric on a high genus surface which is almost flat except for some cone points where all of the negative curvature has been concentrated.