Approval is this module's own workflow, and §9.5 gives its mechanics: who may submit (yourself, for your own time; the submit permission, for others'), who may approve (the project's manager, the employee's supervisor, or a holder of the approve permission), and how a rejection returns to draft with its reason attached. Approve is its own permission, separate from recording and from billing (§1.12) — the three hands on a timesheet need never be the same person.
Where a workflow hands work from one person to the next, the hand-off surfaces through the notification bell and the activity record, as everywhere in the application (§1.7). The working tool for chasing, though, is a report rather than a bell: Timesheet Status (§9.9) shows, person by person and week by week, whose time is approved, whose is waiting, and whose has not been written at all.