Welcome to Android Central! A major system update doesn't necessarily require Google Play Services to update first. Google Play Services should auto-update in the background (although sometimes you'll get a prompt that certain Google apps need Google Play Services to be up to date, in which case tapping that notification brings you to the Google Play Store page for Play Services). Once Google Play Services has finished installing, you won't see any notification that it's done.
It sounds like the main concern here is that the phone hasn't updated to Oreo yet. Were you ever on a Developer Preview?
11-07-2017 10:50 PM