The host often places a default “index.html” file in a public folder such as “www” or “public_html”. It’s often in the same folder as your root WordPress install and its “wp-config.php” file. If you find and delete it, this should let your WordPress install provide the homepage. (One other option in this case is to update your server config so it looks for “index.php” before “index.html”, but usually deleting the file is easier.)
If you’re asking specifically about why the Akeeba Backup plugin isn’t working as expected, you should contact their support team directly. They likely have documentation on what to do, and how to troubleshoot, in addition to support channels.
If you’re asking more generally about moving your site, there are a few things that affect any migration. Number one, there’s DNS propagation time. When you change your DNS settings at your registrar to point no longer to the old host, but now to the new host, that takes awhile to take effect. And if you’re moving to a completely different domain, it takes time for that new domain to propagate.
Number two, you also have to make sure all the settings are correct. It’s not unusual to have to change wp-config.php settings when you switch hosts, so you’re pointing to the correct database with the right username and password. If you’re also switching domains, you’ll also need to run a database migration plugin, as the URL gets so embedded in the database in serialized ways that it usually doesn’t work to just try to update it manually.
So it would make sense as a first step to contact the specific plugin author and ask them for tips. It may also make sense to reach out to your hosting team to make sure you have all the settings correct. The “coming soon” page sounds more like your DNS settings haven’t propagated yet, or perhaps the new host hasn’t finalized all the account setup yet.