Expanding on Mutant Bob's answer (he provided the code, I'm commenting from the math point of view):
It is, unfortunately, impossible "in general" (except some fringe cases) to recreate the real positions of some points in space from how they appear from one point of view (and your scene essentially represents one point of view - from the front - even though it's technically 3-dimensional). You need either two points of view (two different variants of the scene; one would look as if you're observing it from the front, like you have here, and another would look as if you're observing it from the side, for example); or you need one point of view AND a depth buffer for that point of view.
Suppose you have the first situation (you have two points of view). Then the task of recreating the original positions of vertices in space would be pretty demanding (depending on whether or not we can biject vertices from one scene to another easily), but theoretically possible to pull off.
Suppose you have the second situation (one point of view and the depth buffer info). Then the task of recreating the original positions is fairly trivial (you might get lost in the equations, but the algorithm itself is pretty straightforward).
But: unfortunately, even though you have a 3D scene, it's essentially "2D": you only have the "screen positions" of vertices stored in there and placed onto some plane that is parallel to the camera screen. As a result, you have only one point of view and that's all; no second point of view, no depth information.
Here's an example: see that lever on the wall behind the character? It could be a big lever far behind the character, or it could be a small lever right next to the character. Both of these situations would result in the same projection (same image). Thus, it's not possible to "backtrack" this operation and determine where that lever really should be.
The opposite operation to what you want to do is absolutely possible to perform, but what you're asking for is, sadly, impossible since the depth information is lost.
However! If your scene isn't completely flat (all vertices being on one plane), then you could, in theory, multiply it by an arbitrary factor along the axis of depth and then multiply the coordinates along the two other axes, the further you get from the camera position, the more the multiplication factor. However, it will be a very approximate solution (you'll have to eyeball it) and it will largely depend on how (if at all) the relative depth is preserved in your scene.