As tension is increased, almost all energy will be stored in the threaded bar and the elastic properties of the testing machine's framework. Very little energy will be stored as axial compression in the nut.
If the thread on the nut and/or bar strips due to tension, the failure will usually be a slow creep and not shoot-off fast. The stored energy is expected to go into deforming the contact material. The situation will be different with glass or metals that are not ductile.
Integrate the stress strain diagram to find the total energy stored in the threaded bar. That energy is stored evenly along the bar. If the bar then fails suddenly, the proportion of the length of elastic bar still attached to the nut will accelerate towards the nut, then carry the mass of the bar and nut from the machine. The kinetic energy of the ejected mass will be slightly less that the potential energy stored as tension in that part of the elastic rod.
What material is the threaded rod and nut ? Is it ductile or brittle ?