What is the flowing current through an inductor when the switching gets delayed?
I wonder if there is an amount of time delay in switching, what is the
  current through the inductor? Does it become zero? Where does the
  stored energy in the inductor go?


It will instantly produce a spark across the open contact and fizzle the energy to zero pretty much in micro or nano seconds. The inductor does only what it knows as per this formula: -

$$V = L\dfrac{di}{dt}$$

Meaning that the rapid change in current causes a massive voltage and the energy burns off in the resulting spark.