wormholes - how short is a shortcut, really?

[phantom power]

6. The circle is a 1D manifold curved in 2D, so...

in that case a cord is indeed a wormhole cutting through the higher-dimensional 2D curvature.

In our universe, (assuming our universe is hyperpherical), our manifold would be a hypersphere curved in 4D. A wormhole could cut a cord through the 4D curvature. How much distance would this save?

if you take the viewpoint that a wormhole can cut through native curvature, I think the native curvature is very small, and so you aren't going to get a game-changing benefit by cutting through it. If there were people actually bending the universe in tight folds to cut through, I think that would be evident to us. And you'd have to be bending it in various ways all the time to go the places you wanted.