Can I conclude that acceleration happens a bit later after force is felt? We define forces like electric force, magnetic force and gravitational force etc, to be caused by field lines such as electric field, magnetic field and gravitation field respectively. Since these fields take time to reach the object on which the force is applied for acceleration, the acceleration should occur after the force is applied. Also, does it apply to all cases or are there any interactions that happens with contact?
What I think is that when object A applies force on B, A first feels the force and then B feels the force and so accelerates. Means that force applies on B and B accelerates at the same time but A feels force first.
 A: For a particle in a force field (like electric field, magnetic field, gravitational field etc. like you mention) the acceleration of the particle at time $t$ is determined by the value of the field at the particle's position at the same time $t$, in agreement with Newton's laws.
If you change the source of the field (charge a capacitor, move a magnet, etc.) and the source is in another place then you are correct that the field values in the rest of space will take some time to "update". But the particle will always move according to the field strength at it's own position.
A: mathematically  acceleration exists at time = zero, before there is any displacement or velocity since both involve a time integral.
However, because material objects are not infinitely stiff, forces between objects being pressed into contact build according to Hooke's law for elastic solids which means that accelerations in this case are also not instantaneous but newton's laws still hold.
A: For the idealization of point particles, the acceleration of a particle at a particular time is determined by the force at that same instant in time.
This is also true for the idealization of rigid bodies, as acceleration of one point means instantaneous acceleration (in general) of all points in the body in order to keep the points fixed relative to each other.
For non-rigid bodies, it does take a finite time for the "signal" that a force has been applied to one part of the body to propagate to other parts of the body. So in this sense you can think of there being a delay. However, for each point on the body it is still the case that the acceleration is determined by the force at the same instant in time.
