Is it possible to measure the electric current speed through the wire by this method? This method is very simple but I am not sure if it would provide accurate measurements.The current electrons deriving from the battery should have some speed and in that way should interact with the coil shown on the picture by pushing the atoms of the coil lattice until the lattice gains the speed equal to the speed of the electrons so it starts rotating.Of course the contacts of the coil with the wire should be of negligible friction and the coil should be as symmetric as possible to provide neat rotation.If the drift velocity of electrons is let say 0.2 mm/s should then the coil rotate at that velocity? 
 A: The electromagnetic wave propagates through the wire at the speed of light. Not exactly, but for purposes of this experiment you can think of this to be true. It is no less than half of the speed of light.
Our macroscopic understanding of current is via flow of electrons. But electrons do not actually travel down the wire. They do move, but at a drift speed on the order of 1 mm/s. When we press the light switch, it does not take minutes for light bulb to turn on or off. In other words, electron flow is not the main mechanism which is responsible for power (signal) transmission.
Having this in mind, what exactly do you want to measure?
A: A back of the envelope in terms of momentum conservation calculation might indicate the difficulty of observing an effect?
The current is switched on and free elections in the copper wire, starting from rest, drift at your estimate of $0.2\,\rm mm \, s^{-1}$.
To conserve momentum the copper lattice starts to drift in the opposite direction at a speed $v$.
Putting in some numbers, for the relative atomic mass of copper $(63.5)$ and that of an electron $(1/1840)$, and assuming momentum conservation gives, $\dfrac{1}{1840} \times 0.2 \times 10^{-3} = 63.5 \times v \Rightarrow \mathbf {v\approx 2\times 10^{-9} \,\rm m \, s^{-1}}$.
