It all starts at the top. Convince your CEO, department managers, or whomever drives the agenda that they will derive positive benefits from the process, and they will then become your greatest champion. Other managers will ignore them at their own peril.

So how do you convince them?
1) If your process does not do this already, make sure that reviews are tied to goals. Each staff person's goals should directly support the manager's goals. If the manager knows that they will be judged/compensated based on accomplishment of their own goals, then they will want to regularly review their staff's performance to ensure that the staff is working to make them look good. This is about improving the bottom line - both the company's and the manager's.
2) Hiring is a pain. And poor retention leads to more hiring. If a manager has been losing staff, demonstrate to them how the review process can resolve that problem. There are lots of surveys showing that people leave jobs because they feel unappreciated and/or unguided. The review process is a great opportunity to tell/show people that you appreciate their efforts, and it is a time to lay out a plan for that individual so they feel that their days will have meaning.

These two approaches won't work in all cases, but they're strong reasons to do performance reviews - and they have worked for us in getting managers to complete reviews on time. Good luck!

As a practical matter of simply getting them back, you could post the evaluation "schedule" that shows the employees exactly who has the evaluation at any given time. If it's not turned in on time, an email could go to that entire department saying the submittal deadline has been missed and therefore raises cannot appear on the next paycheck. Let the employees hound the manager for you. Similarly you could alert the other managers who is holding up the entire process.