When you want to work timesheet using Project Manager Approval Process there are some things that is good to consider in order to work with this feature.
All CA PPM Versions.
Permissions: You don't need timesheet permission to work with this, since the process send the action item and you can approve from there.
Process Properties: In Start Option tab, you need to change start option to Auto-start and start event to update, if you don't put Auto-start the process will never run.
Resource Manager and Project Manager Timesheet Approval Rights
Is the Project Manager automatically given security rights to view and approve the Timesheets?