If your restore is stuck at 0% for more than 5 minutes that means there is something in your servers configuration which is blocking Backuply from running the Restore.
The most probable configuration could be the .htaccess rules to prevent execution of PHP. The rule to prevent execution would look something like shown below.
Deny from all
Note: The rule in your .htaccess might not exactly match with the one giving here but will be similar.
So now what you need to do is you will need to whitelist 2 files which Backuply uses to restore a website the file are restore_ins.php and status_logs.php
To whitelist the files you will need to add the rule given below just below the rule you found to block the execution of the PHP file.
Allow from all
Note: If you are not aware of how .htaccess works or you are not sure what you are doing then its better to ask your host or contact us to do it. Because, if you end up doing something wrong in the .htaccess file then it will take your site down.
Once you add these rules and save it then Backuply restore will work. Make sure you check the .htaccess files inside /wp-content and wp-content/plugins directories too, because sometimes block rules are used there too, so you will need to add the rules in all 3 locations.
If you face any issue you can always contact us at firstname.lastname@example.org