Does NTIME($n^\alpha$) $\subset$ EXPTIME imply NP $\subset$ EXPTIME?
Upon reflection this is an easy consequence of the Time Hierarchy Theorem.


NTIME$(n^\alpha) \subseteq$ TIME$\left(2^{n^\alpha}\right)$ by brute force
TIME$\left(2^{n^\alpha}\right) \subset$ TIME$\left(2^{n^{2\alpha}}\right)$ by the Time Hierarchy Theorem
TIME$\left(2^{n^{2\alpha}}\right) \subseteq$ EXPTIME obviously


So there's nothing new here.