First I should tell you when NOT to introduce them, and that's while the student is still a beginner. The beginner, adult or child, is focused on much more important things in the first months (first years, even, for an early elementary child). What is completely occupying the beginner's attention is how to get the right finger to the right location on the keyboard at the right time and how long to hold it there while preparing to do the same thing on the next note. When you stop to think about it, that's a whole lot to keep track of!
I am certain you were brought up on a "diet" of scales as a student. I know I was. It was obvious to me that these patterns were not in any of my beginner songs, so I chose not to work on my scales because I recognized this and also needed the time to work on the songs [which didn't have them]! My teacher used to give me letter grades on my scales. (What a stupid idea! I recognized that early on, too.) You will not be surprised to hear that I got grades of D and D+ on scales every week. And I didn't care! He tch-tch'ed at me and talked to my parents about what a rebel I was, but I held my ground. Doggone it! Scales were dumb. And I didn't give a fiddle what grade he gave me on them. I'm still not a big fan of diatonic scales as a technical study, but, yes, later I did become proficient at them - - when it was obvious to me that I needed these physical movements. (Chromatic scales, on the other hand, are fun to play! Candy for the fingers!)
Ok, enough confessions. But perhaps you had some similar feelings as a student.
So, when is a good time to introduce diatonic scales? I believe that is when the literature calls for them on a frequent basis. Clementi sonatinas come to mind readily here. That means the intermediate level. So here your beginner can go a couple of years and never even worry about diatonic scales. What fun!
As an aside, I am a big fan of the "new" scale fingerings in which the thumbs play concurrently. Makes so much more sense than the "old way." And I also request that my students never use finger 5 in a scale (as a technical study). Yep, start and finish with a thumb, even though it's a bit awkward. Why miss an opportunity to practice a tuck? Time enough in literature to take a shortcut.
One final aside: why must we teach exactly the way we were taught? I decided long ago (guess with which teacher?!) that certain things didn't make good sense and I wasn't going to do them this way "when I was a grown-up."