Past exam of the options II course of the University of Oxford / 2013 / O4 / d / Solution

Yes, it is a uniformly most powerful test for the one-sided alternative. For every fixed ${\sigma }_1^2>{\sigma }_0^2$, the likelihood ratio is increasing in the same statistic $S$. The size-$\alpha$ threshold found in part (c) depends only on the null variance and $\alpha$, not on the selected alternative variance. Thus one and the same test is most powerful against each simple member of the composite alternative, which is exactly uniform most powerfulness.