Email notifications for timesheets, out of the box without using a process, is purely driven by timesheet approval right.
The global right should be avoided unless someone in the organization needs to get a notification from every timesheet submitter; this right also has been found to work inconsistently on certain versions.
Try to set up (approver to submitter) instance to OBS, or OBS to OBS rights so that a specific user has the right over everyone on an OBS (and descendant if desired); or everyone on an OBS has the right over everyone on another OBS (and descendants if desired).
Whoever has the timesheet approval right over another user, will get an email when that user submits a timesheet. But if you want to have email sent NOT based on this access right but resource/project hierarchy and properties, such as based on who the Resource Manager of the submitter is, you would need a process.