Is force due to electric field instantaneous? If not then why? Suppose a charge $q$ is experiencing a force due to charge $Q$. Suppose we move the charge $Q$ very slowly (no acceleration) what's the instantaneous impact on the charge $q$? How will the $q$ react? 
 A: If you accelerate Q, then you will create a changing electric field, which creates a changing magnetic field, which create a changing electric field, etc., and the result will be EM waves emanating from Q, and they will travel at c to q, changing the force on q when they arrive.
But if there's no acceleration, there's nothing to react to. SR says that there is no such thing as an object moving in an objective sense; there is some frame of reference in which Q is stationary and the electric field is fixed. q will experience a changing electric force as it moves through different parts of that field, but the field itself remain constant, and there is nothing to propagate.
A: No two things in the universe happen "instantaneously", unless they are at exactly the same location, because "instantaneously" would have different meanings for observers moving at different velocities.  Maxwell's equations, which describe electromagnetic interactions perfectly for most practical purposes, contain time-dependent terms that describe the propagation of changes in an electromagnetic field.  If your Q is moved at all, whether fast or slow, the resulting change in its field at a distance D does not occur until a time t = D/c.  That is, the change propagates out from Q at the speed of light.  This is an observable fact that, per special relativity, is the same for all observers.
A: It is not instantaneous.According to Lienard Wiechert potential, where one can see that the effect of the charge does not travel faster than light.So force doesn't acts instantly. 
A: Suppose you consider q to be at rest.  If Q is moving, then the electric force felt by q※ will point to the delayed position of Q.
If Q is moving uniformly in a straight line, the actual acceleration felt by q, by strain gauges arranged around it, will point in a different position — to the instantaneous position of Q in your reference frame!
At least, that will be the equilibrium state some time after Q starts moving at a steady pace.
This is due to magnetic effects of the moving charge.  At least, that is how you interpret the full 4-dimentional majesty of the electromagnetic force from an inertial reference frame.

※  How do you measure just the pure electric force?  A measuring device might be a small dipole pointer.  But that is made using charges on one or both ends, which feel the same secondary effects.  Off hand, I think the “true” pure-electric charge cannot be measured locally, since the interpretation of the correction effects depends on your reference frame.
But if you start with stationary charges, and monitor everything when Q starts to move, change direction, and other experiments; you can see the initial delay of any effect and work out the rules consistent with the finite speed of propagation.
A: You can treat that as an instant if you are working with small enough distances and velocities, but it's not. If you'll ever study field theory you'll meet retarded potentials that are just this: the field propagates at the speed of light and it's no longer seen as instant
A: Electric force is instantaneous in Newtonian physics. In general relativity it is not. 
