The center of mass of the block DOES accelerate "immediately".
Those molecules closer to the point of application of the force have a
much larger acceleration than the remainder of the block, immediately
after the force is applied. The macroscopic view of the block will not
show evidence of this acceleration until the internal deformations
propagate through the block.
Newton's Second Law refers to "particles". Application to microscopic
views of complex systems require other relationships.
Indeed if
one thinks about what happens in a microscopic, short time frame the "push"
on a block does not immediately accelerate the whole block. The information
about the push must propagate at a speed slower than c before the rest of
the block begins to accelerate. The actual speed of propagation would be
the sound velocity.