If the exotic matter were suddenly removed from a stable wormhole that is spherically symmetrical and nonrotating, it would become a black hole. This immediately follows from Birkhoff's Theorem -- the only spherically symmetrical solution to the Einstein field equations in matter-free space is the Schwarzschild (non-rotating Black hole) solution.
Observers on both sides of the wormhole would note the disappearance behind an event horizon of that part of the wire nearest to the wormhole's throat.
To observers stationary relative to the newly formed black hole, the black hole would appear frozen (as is well known) with the wire jutting out of the event horizon.
To observers suitably accelerating toward the black hole, the black hole would appear to pinch off (as is well known) with the wire being severed.
The only proviso in this explanation is that the presence of the wire makes spherical symmetry only approximate. I have assumed, perhaps incorrectly, that the solution perturbed by the presence of the wire is essentially the same as that in the wire's absence.
Sources: Gravitation by C. Misner, K. Thorne, & J. Wheeler (1973), The Physics of Stargates --Parallel Universes, Time Travel and the Enigma of Wormhole Physics by E. Rodrigo (2010)
Minor quibble: Matter's violation of the Dominant Energy Condition is not sufficient to support a stable wormhole. For such a wormhole to allow the traversal of material objects, its matter must violate the Weak Energy Condition. [In other words, it's possible to have matter that violates the DEC that is insufficiently exotic to support a wormhole.]