Deleted important converstaion

iOS 16 has a Recently Deleted option that would allow a user to retrieve a message that was deleted in the last few weeks.

But......you would have to have installed iOS 16 on the phone........before......you deleted the message.

Delete and recover messages on your iPhone or iPadOfficial Apple Support › en-us

If you are running iOS 15:

When you say you haven't set up iCloud 'backup' for messages...

Do you mean that in Settings > [your name] > iCloud > Messages, that switch is OFF?

If so.. that switch is NOT 'iCloud backup'.. it's syncing.

But in any case, there is a chance at least.. you would need an iPhone backup that contains the messages.

Keep in mind that a new backup generally replaces the prior backup.. So if you deleted the conversation 3 days ago and your iPhone backup is from 1 day ago, the messages are not going to be part of that backup

Locate backups of your iPhone, iPad, and iPod touch - Apple Support

You would need to restore your iPhone with that backup, which means you will likely lose other stuff between now and whenever your backup occurred.

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

If you are using iOS 16:

and... you deleted the conversation while using iOS 16.. within the last 30 days or so..

Delete and recover messages on your iPhone or iPad - Apple Support

also, a good user tip from one of the community members here

Recovering deleted texts and iMessages - Apple Community