I would tie it to the manager's performance review. If the manager doesn't complete the reviews of his/her subordinates on time then they don't get a good rating on this key job responsibility, which could ultimately effect the salary increase this person would have gotten.
You can also start disciplining the managers over this. It may seem harsh, but if you start doing this and ultimately have to fire a manager over not doing a part of his/her job you would get your point across to everyone.