I just had similar Wi-Fi connection issue plus also losing connections to my remote nest temperature sensors.
I was able to resolve the issue by resetting my Nest Thermostat to factory default setting. The software version remained at latest 6.2-27. after factory default.
I took the followings steps:
1. Removed the devices from my Nest account. In my case using iOS Nest app.
2. While Nest Thermostat is still plugged in and powered, dial to Settings - Nest App - Reset - Factory Default
3. After reset, using iOS Nest App, follow the steps as if adding a new Nest device.
Note: Unfortunately, this will reset Nest past temperature, it’s e-Learning histories, and need to redo schedule. However, a small inconvenience when comparing to having only a hockey puck on the wall.
So far, so good for me.
6.22-27 update is absolutely a thermostat killer. Since update, BOTH therms will NOT stay connected to Wi-Fi. The App will not respond and sync to the therms. I spent 4+ hours uninstalling and reinstalling both therms and the software; then spent 4+ hours on 2 calls to ‘customer service’ who can see the issue , only to be told I need to buy new Displays at full price to fix. No way. It’s a software/ firmware issue but Google/Nest won’t acknowledge it. I’m moving on to a better company.