The place where you cut the wire acts as a temporary Capacitor where a huge potential difference is formed. This potential difference causes an intense electric field to develop, which is where the energy is initially stored. If the potential difference developed exceeds the dielectric breakdown voltage of the intervening medium, the charges are lost as a spark discharge which dissipates energy as EM waves and heat.
But usually the current is never cut down this abruptly, providing enough time for the energy to dissipate as the normal safe resistive heating. If not, then the energy will be lost by the aforementioned discharge which is very intense and might damage the equipment under consideration. Hence the use of a parallel capacitor with a large inductor, which allows slow dissipation of energy as LC oscillations (EM waves) and normal resistive heating.
The said capacitance ceases to exist only if a spark discharge dissipates the gathered charge or, the instantaneous back emf is slowly reduced by resistive heating (the circuit is not cut-off). (i.e. if we assume the cessation of current occurred instantaneously, the developed field would exceed the breakdown field leading to a spark, or if we assume that the change is slow enough so that no spark is developed, then the finite time it takes for the current to die down, the resistances of the circuit would dissipate the energy in that time). The sudden stopping of the current is only an ideal occurrence and does not occur in practice.