Layouts can be exported and imported in and out of the Report Writer. Clicking the 'Report Tasks' button at the top-left of the report, there are two options. 'Export layout to file...' and 'Import layout from file...'.
'Export layout to file' will open the windows explorer where you can select a folder in which to save the .repx file (Report Layout file). Click 'Save' to export the Layout.
'Import Layout from file' will open the windows explorer where you can search for the required .repx file (Layout file). Select the file you wish to import and click 'Open' to import.