This is one of those "yes ... and then again, no" things.
It depends on the granularity that you are viewing the process.
Classically there is no time lag, though there will be a gradual increase in the acceleration right at the start and classically is where we live most of the time.
You can get a lag if you apply a quick force against something squashy or springy - but then the immediate effect is deformation of the body ... which is local, rather than bulk, motion.
Then there is the illusiary lag due to the initial motion being too slow to notice ... similar to hang time.
Did you have a particular context in mind?