Any updates to the Wordpress installations using the wp-admin UI feature are prohibited. The site owner (You) own the files, and the web server software is not allowed to update any files on the site.
We run a backup nightly on all files and databases. That takes the time somewhere between 01 AM - 05 AM UTC every day.
After a successful backup we launch an update checker impersonating as the site owner (You) to see if there is any plugins needing updates, and process the updates automatically. This process is automatic for all site installations, but can be disabled by a request by the site owner (You).
It is also possible to update Wordpress (and do other maintenance) using the tool wp-cli.
You can access wp-cli from SSH-shell.
$ wp help
To list all plugins you can issue command:
$ wp plugins list
To update all available plugins at once:
$ wp plugins update --all
You have full and unlimited access to your files. Only the Web UI is restricted, as that is the main attack vector for Wordpress sites.