As of September 2018, saves can be backed up to the cloud, and restored without the original Switch. However, you must pay for online service to use this feature, and you must have purchased online service before losing access to your old Switch.
According to Nintendo's product page, backups are automatically taken whenever you have an internet connection (and you have paid for online service). Their help page describes the following procedure to download cloud saves onto a new console:
- To download a backup from the software menu:
- Open the game's software menu and select "Save Data Cloud Backup."
- Select the desired user, then select "Download Save Data."
- To download a backup from within System Settings:
- From the HOME Menu, select "System Settings" > "Data Management" > "Save Data Cloud Backup."
- Select the desired user, then choose the backup file you wish to download.
- Select “Download Save Data.”
The page also warns that this procedure will erase any local saves. The same settings page can also be used to verify that your saves have backed up successfully, if for some reason you're concerned about that.