Google made clear that it would apply the Nest name to all its smart home products, but just how would it rename classic products? You now have a better idea. As of this writing, a visit to the Connected Home Accessories section of the Google Store's Special Offers page shows the Google Home's replaceable base as "Nest Home Base." The product page still refers to the speaker by its original name, but you'll find this change in the terms and conditions as well.
We've asked Google for comment. It's not certain if an official renaming is right around the corner, especially when the Home Mini and Home Max aren't affected yet. However, it seems like changes are at least underway. This also suggests that Google isn't waiting for a hardware refresh to apply the Nest badge to one of its most iconic products -- you may see the Nest name everywhere even if the devices are otherwise the same as before.
Update: Android Police heard from a spokesperson who called the change a "mistake" and said it is in the process of being fixed.