Recovering a point embedding from a graph with edges weighted by point distance
One algorithmic approach to solving this problem: treat this as a set of nodes, connected by springs, then let them settle/relax into shape.

Each edge $(v,w)$ corresponds to a spring; if the distance between points $v$ and $w$ is supposed to be $d_{v,w}$, then the spring is chosen so it ideally wants to be of length $d_{v,w}$ (it can be longer or shorter, but this costs energy).  We now want to solve for a set of positions that minimize the total energy.  Suppose each vertex $v$ is placed at point $x_v \in \mathbb{R}^3$.  Then the total energy of this arrangement will be

$$E(x) = \sum_{(v,w) \in E} (\text{distance}(x_v,x_w) - d_{v,w})^2.$$

Here the $d_{v,w}$'s are given (they are the weights on the edges), and we want to solve for the $x_v$'s (they are the coordinates of the points).

We can solve for an arrangement $x$ that minimizes this total energy.  This arrangement then provides a reasonable candidate for the positions of the points.  This is an optimization problem, and there are standard techniques for solving this kind of optimization problem.  See, e.g., the article Network Solutions by Erica Klarreich.

I don't think there any guarantee this will provide the correct desired solution.  It's possible that the optimization problem might settle to a different optimum that doesn't reflect the actual arrangement of points you were looking for.  However, if your graph is sufficiently dense, I suspect it might often work and give you the desired solution.



Footnote: Of course even in the best case we can only solve this problem up to translation, rotation, and reflection since those transformations preserve all distances.  Thus, you can't expect a unique solution -- but you might hope that the solution is unique up to translation, rotation, and reflection.



Finally, there's a lot of work on embedding graphs into $\ell_2$ space, while minimizing the distortion of the embedding.  That's very related; you are basically asking for a zero-distortion embedding into $\ell_2$.  Thus, the techniques developed in that context might be useful for your problem, too.  Typically, that work focuses on finding a low-distortion embedding, because that work focuses on the case where there is no perfect embedding that makes all the distances match exactly, so instead it looks for a low-distortion solution (one where most edge distances match pretty well) -- so that work is focused on a slightly different problem.  However, it's possible that their techniques might be effective in your situation as well.  It's worth a try.