Why does NTIME consider the length of the longest computation?
Because you don't know ahead of time whether or not any given input is a 'yes' instance &mdash; that is, whether there exists any accepting path &mdash; it makes sense for the sake of uniformity to bound the run-time independently of any particular feature of the computational paths. Thus, it makes sense to require the worst-case behaviour to be polynomial time, regardless of whether or not any accepting paths exist.