Go to JOSM's [imagery preferences](https://josm.openstreetmap.de/wiki/Help/Preferences/Imagery), imagery preferences, go to the "Cache" ["Cache contents"](https://josm.openstreetmap.de/wiki/Help/Preferences/Imagery#Cachecontents) tab, and use the "delete" "Clear" button in the relevant line.
The tiles for the JOSM download dialog window can be deleted by using the "delete" button in the "*Mapnik*" [the "*Mapnik*"](https://josm.openstreetmap.de/ticket/17054) line (needs restarting afterwards).
Technical details (and the hard way of deletion): The tiles are saved in `~/.cache/JOSM/tiles`. For me exactly in the files `TMS_BLOCK_v2.data` and `TMS_BLOCK_v2.key` (stored with the key "Mapnik" if I see correctly). Deletion of the whole directory or just both files results in JOSM downloading the tiles when opening the download dialog.
Tested with version 14460 on Linux. For cache paths of other operating systems see [JOSM help](https://josm.openstreetmap.de/wiki/Help/Preferences#JOSMpreferencedatacachedirectories).