Oh, the 'joys' of working with a shared system and no terminal access! (NOT!)
Automated system is probably saving the backup files in the sea of files of moodledata/filedir/ ... while there is a way to find those .mbz files via query of DB (contenthash) there's the issue of using wget to acquire them by humanly recognizable names! Too much!
So ... suggestion ... using cPanels file browser, create a directory called 'cbu' and give it permissions to be seen by apache user.
Reset automated backups to save to a designated directory ... that of /home/userlogin/public_html/cbu/
In moodle admin interface, go to /admin/tool/task/scheduledtasks.php ... if you have the path to PHPCLI set in paths of site, you will find a 'run now' link in the row for Automated backups. Run Now might time out .. limits of shared hosting.
IF it does complete, then the wget URL's would change to
You could then create a 'getlist' file in the file system repo of new server that consist of one liners ... url's to each backup file.
Example - each reference on a single line no extra lines:
Then, using wget in the file system repo:
wget -i getlist
command above reads the 'getlist' file 1 url at a time and acquires that file, then 'gets' the next.
XFER is server to server thus one is working at faster speeds and hopefully won't take long - depending upon number of course backups and their sizes.