Export link
It is now possible to download a report by a link, e.g. https://YOURDOMAIN.atlassian.net/plugins/servlet/ac/jira-timesheet-plugin/timereports#!pivotTableType=Timesheet&export=xls for Timesheet Report and Gadgets or https://YOURDOMAIN.atlassian.net/plugins/servlet/ac/jira-timereports/timereports#!pivotTableType=Timesheet&export=xls for Time Reports.
All parameters from url will be applied, so please set up report needed, copy url then add '&export=xls' to the end and share url received.
Such link might be used in Confluence page for example.
Please note, the ‘export=xls’ is url parameter and exports 'exportData.xls'. It is possible to export ‘exportData.csv' (export=csv) and 'exportView.xls’ (export=html).
Please note also, report window will be opened before download.