If a non-deterministic Turing machine can just "guess" the correct answer to a problem, does it do this in constant time/immediately? Also, does this also apply to problems in NP too?
A nondeterministic Turing machine can guess at most one bit each step — this is how they are defined. In particular, a nondeterministic Turing machine running in constant time can be converted to a deterministic machine also running in constant time. This should allow you to answer your own question.