In every timesheet (User, Project, Team, Account, Advanced) is the possibility to choose "Report". This gives you a report view of the timesheet and gives you access to the Excel eport. Please take a look at https://tempoplugin.jira.com/wiki/display/TEMPO/Reports for more information.
All reports can be exported to Excel directly from Tempo report view. This blog explains how you can export and pivot your data in excel
You can also use the Tempo API to export all data on a high level permissions as documented on this page:
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