Is EMF generated instantaneously? I know that EMF is induced when magnetic flux changes over time. But is the production of EMF an instantaneous process or is there a delay in its production? 
Consider the example below.

By moving the magnet towards the ring I increase the flux. By Lenz's law the current should be induced such that it opposes the change in flux and EMF is induced. But does the process of changing the flux and production of EMF happen at the same time or is there a delay?
An analogous question: is there a delay in production of EMF in an inductor as its current changes? Consider the LR circuit below:

The inductor produces back-EMF. According to my teacher, the production of this EMF is done instantaneously(in the same time as the current is changed, further he supported this by giving the analogy of force and production of velocity) whereas I wonder if there is a delay and if the back-EMF produced at $t=1$ is due to the change in current at $t=0$. 
Essentially, I don't understand how something can happen instantaneously. I expect there should be a delay in the cause and the effect. It feels like Zeno's dichotomy paradox. I could keep on dividing the time until I reach a point where I could distinguish the cases (here, cause and effect). 
I also don't understand the in-depth process behind Faraday's Law and Lenz's Law. I urge you to recommend to me any content that explains "why" this happens. 
 A: Let's consider a simpler question. Consider Newton's second law,
$$F = ma.$$
When a force is applied, does acceleration happen instantaneously? According to this law, yes, since they're always proportional.
You're right to point out that this seems in contradiction with relativity. The resolution is that the force isn't applied instantaneously. All realistic forces are continuous in time, have correct relativistic propagation delays, etc.. That makes accelerations satisfy the same property.
Now we turn to your question,
$$\mathcal{E} = -\frac{d \Phi_B}{dt}.$$
Since this is just an equality, a changing magnetic flux always instantaneously creates an emf. But again, this isn't actually instantaneous, because changing magnetic flux itself takes time! 
In particular, if you do a detailed calculation, you can show that immediately after switching on an inductor, the magnetic flux in any surface surrounding the inductor is zero. It only becomes nonzero after a finite time, due to the relativistic propagation delay of the electric and magnetic fields. The exact way this process works is explained in my answer here.
To directly answer your question involving the LR circuit: if we switch on the current at $t = 0$, then at $t = \epsilon$, after a finite propagation delay, both the EMF and magnetic flux switch on at once.
