As a user without root, you can't restore a full backup or run the "copy account" WHM operations, so basically you're stuck with a very time consuming and error-prone restore process.
In most cases, your webhosting company will do an account copy, or restore a full backup, without charge, so the fastest way would be to supply them with all the details in one hit and ask them to do the copy. You need to give them your reseller name, and the URL, the username and password for the account to be copied. They'll need to change the reseller name to yours after uploading the account to your server, remind them if they forget. You'll need to coordinate with them to change the nameservers after the copy - also change the MX record to point to a domain on the new server as soon as the copy is complete to avoid email going to the old server and getting lost.