How can I recover deleted text messages not backed up in iCloud

You cannot. There is nowhere to recover them from. The only backups of messages are the ones you make. If using messages in iCloud, deletions are permanent so it is up to you to archive messages you need to keep somewhere. Many people do that by transferring messages to their computer using software on their computer (e.g. PhoneView or iExplorer).

But messages are not like email. There is no archive or trash folder and text service providers don’t keep any copies anywhere - they purge content the moment it is delivered to the recipient (or with a short few days if unable to deliver a message).

Unless you have a local backup, you can't.

It's gone, deleted.