Why do we get a shock from electric wire instantly? As we know the drift velocity of electrons in conductors is very low and it is about a few millimetres per second. Now suppose I hold one end(naked) of a wire which is not connected to any source. Then if we connect the other end of the wire to an electric board and then if we turn on the switch, we instantly get a shock.
But the drift velocity of electrons is very low so how is it possible that we instantly get a shock after turning on the switch ? We should get the shock after some time because the electrons will require some time to reach the other end of the wire.
 A: When current start flowing in a conductor, the electron doesn't have to move all the way from one end to the another.
The electron at every point of the conductor start moving at the same time, and causes current to flow.
This means that the end of conductor you are holding, in that end point the electrons start to flow to your hand as soon as the switch is closed, and thus you instantly get a shock.
Here is a rough diagram I made for you,

A: What is "instantly"?
Human reaction time is about a quarter of a second, so everything that happens faster than that is perceived as instantaneous.
Why the drift velocity is not the limit
Electrons do not have to flow from one end of the body to the other for electric current to occur. In fact, even in an electric circuit they rarely travel very far. The force exerted by electrons on each other however propagates much faster.
Yet, it is not anywhere close to the speed of light
It would be incorrect to say that it propagates as fast as in a conductor, which is a crystalline solid: human body is largely an amorphous material and most of the conductance occurs via electrolytic reactions, i.e., movement of ions in the body cells: thus, the worst case scenario is that the velocity of ions and electrons need be compared with the cell size.
Furthermore, the effect of the electric field/current needs to be translated into a physiological response - mostly in muscle contraction in case of an electric shock.
Additional reading:

*

*see here for basic background on the effects of electricity on human body

*here for the basics about functioning of muscles

*here and here for the effect of electricity on a muscle

A: The wave through the electrons is what matters, not the movement of the individual electrons. The wave is what carries the energy. This is similar to the difference between the speed of sound and the much slower speed of wind.
The 50Hz/60Hz current from the powerplant doesn't mean each individual electron is traveling back and forth between your house and the power plant 50/60 times per second regardless of how far your house and power plant are.
A: Think of a similar circuit, but made out of pipes and water instead of wires and electricity. There is a pump some distance away. When the pump is turned on, water starts flowing slowly from the pump, through the pipes, and back to the pump.
Not everything travels at the same speed in the circuit. Tap on the pipe, and a sound wave propagates through the water, whether the water is moving or not. The sound wave travels at the speed of sound, which is much faster than water travels.

So how long does it take for water to start moving at the end of the circuit where you are?
The easy answer is instantly. All the water has to start moving at the same time. The pipes are full, The pump can't force new water into the pipes until some water leaves the other end and all the water inside moves down.
A more careful answer would be that water doesn't instantly reach fully speed. The pump exerts a force that accelerates it.
But also water is very slightly compressible. So a tiny amount of new water can be forced into the pipe before the water downstream starts moving. The compressed water near the pump pushes on the uncompressed water next to it. That water gets compressed, and pushes on the water next to it. This should remind you of sound. The compression propagates at the speed of sound in water.

The speed of electricity is more like the speed of sound wave than the speed of the flowing of water. In pipes, you are interested in getting water out. In an electric circuit you are interested in getting energy out. Energy is stored in a region of slightly compressed electrons.
In AC, a generator pushes electrons back and forth at 60 Hz. The electrons vibrate more or less in place. The compression wave travels down the wire, causing electrons all along it to vibrate. You might expect the compression wave to travel at the speed of light, but it is actually somewhat less because of how electrons interact with the atoms in the wire.
For DC, it is much the same, except the battery pushes the electrons one way. A compression wave propagates down the wire, and very quickly reaches the end you are interested in. Electrons begin drifting in one direction. But you don't have to wait for an electron the battery puts in to reach you. You have a region of compressed electrons as the wave reaches you. You can harvest the energy by letting the compression relax.

Usually one does not focus on the compression of electrons. Instead, one talks about it in a somewhat disguised form. A Volt is a Joule per Coulomb.
It is much like water and pipes, where one doesn't talk about the compression of water and how it relates to water pressure. One just talks about the water pressure.
A circuit might contain a battery and a resistor. The battery pushes electrons into the wire, slightly compressing them. The compressed region (high voltage region) extends down the wire to the resistor. The resistor impedes the flow. Electrons "squeeze through", and are uncompressed (low voltage) on the other side.
Outside the battery, electrons flow from regions of high compression (high voltage) to low compression (low voltage). Voltage differences are important to circuits.
