I think that gamification would help as well as make the timetracking more meaningful.
The first gamification element to consider is feedback. Let the employee know their input performance right away (i.e.: how many end-of-day updates they missed/made).
Follow that up with leaderboards. Make it public who is more successful at making their targets (top n teammembers; I don't think its useful to humble the bottom n). In a large organization you may need to limit the leaderboards to departments so that the comparisons are more meaningful to the participants. Plus it's going to be the department manager who is responsible for improving the participation level. Maybe a department level leaderboard to show which departments have the best participation.
Badges can be issued for improvements/successes in participation. i.e.: successfully updated 5 days in a row, 4 weeks in a row, etc.
You probably should reward participants on a variable reward cycle. So that there is a tangible benefit to performing well in the 'game.' Variable so that there is no feeling of entitlement nor is it boring that 'the' top scorer, who can never be displaced for whatever reason, is always the winner. For example the person who earns the most badges this month gets a designated parking spot for the month.
In that most companies that I've worked for, adhesion to the time tracking polices was part of the job description, a high-level of competency in performing this job duty should be rewarded (higher evaluation during performance appraisal = increased merit raise or bonus). Therefore performing well in the game would also provide a tangible benefit. Performing poorly would lead to increased training, counseling, etc.
As noted in other responses above, gamification is not going to be as helpful if there is no useful purpose to gathering time tracking metrics.