plpbrady - The mobile Work Orders screen wasn't designed that way. It was designed so that one user would use the device for their own team rather than a supervisor for many teams.
It would be possible by customization, though. If you removed the filters on the List grid so that it showed all work orders, not just "my work orders" you would see everyone's jobs. Also, if you put extra actions on the Select link such that it populated the employees on the Time tab based on the team for the job that was selected, then the Start timer would work fine.
The web service would work since it is just accepting values from the employees in the Time tab - so the trick really is to get the right employees listed based on the work order selected.
Certainly possible, but not for the faint of heart when it comes to customization.