Can a language recognized by a NFA be recognized by a push down or Turing machine?
A language is recognized by an NFA iff it is regular. It is recognized by a push-down automaton iff it is context-free. It is recognized by a Turing machine iff it is computable. Every regular language is context-free. Every context-free language is computable.

Even without knowing these facts, it is not difficult to simulate an NFA by a push-down automaton and a DFA by a Turing machine.