Google Nest Wifi is designed for working without the ethernet cable between them, that's the whole point of mesh. 🙂 This is called "wireless backhaul". As long as all the points are made for the same region (don't mix US and EU for instance), it will work just fine.
See following documentation about how to set up Google Nest Wifi:
But if you can use cable, you should, as it will give better speed and stability. This is called "wired backhaul". See more about that here:
I just wanted to follow up real fast to see if you were needing any additional support or to see if you have the answer and solution you needed. If you still need any help, just let me know and I'll be happy to jump in. Big thanks to olavrb and Ashepherdon for their info and help.