Is a wormhole detour in spacetime really shorter in distance or time than a straight line?

While any $n$-dimensional manifold can be embedded in $R^m$ for some $m>n$ that doesn't mean the metric in the embedding space corresponds to the metric on the manifold. Of course, usually we do embeddings so the induced metric corresponds to the intended metric for the manifold but this is not always feasible.

One wormhole manifold that is allowed by GR (by assuming the existence of some exotic matter to make the field equations match up) is flat space with two spherical volumes excised and their surfaces identified. The hand-wavy exotic matter is assumed to be on these surfaces (any argument that you cannot get this matter requires some argument from outside GR, like the energy conditions). This is a valid solution, and a path through the wormhole can clearly be shorter than a classical path. Note that this manifold will require a fairly high-dimensional space for embedding, and will not inherit its metric.

Whether physically possible wormholes are (1) possible, and (2) shortcuts depends on the actual matter fields available (and what rules there are for topology-change). The classical wormhole papers assumed exotic matter that enabled shortcuts, but it is not clear such exotic matter exists (or not). Some recent papers with no shortcut wormholes seem to be based on more realistic (?) matter fields. But it is not clear what the actual answer is yet.