After Google officially introduced the new Gmail, they put an approximate timeline on when G Suite users would have no choice but to use it: "about 12 weeks" after everybody got the option to opt into it. The company has now announced that beginning on October 16, G Suite users will no longer be able to opt out of the new experience, and shortly thereafter, they'll be automatically transitioned to it.
By default, users at organizations that didn't opt in to early access of the new Gmail will start seeing the option to try the refreshed layout beginning on August 21, and their accounts will switch over automatically beginning on September 18. Between then and October 16, they'll be able to opt out to return to the old layout, but after that, it's the new Gmail or nothing.
Administrators are able to speed up the process by either granting users the ability to opt in immediately or just opting in for them, but in either case, they'll still be able to opt out until October 16. After that, they'll be migrated automatically.
While this applies to G Suite users, specifically, it's a good indication that it probably won't be too long until the general public loses access to the old Gmail, too.
- G Suite Updates Blog