The problem here is Nest Mini, and other Nest devices, simply don't support these kinds of Wi-Fi networks, known as captive portals. Even if you managed to connect it in the past, in most cases the Wi-Fi provider enables a feature called client isolation, which is a security feature designed to prevent devices connected to the same Wi-Fi from communicating with each other (i.e. your Nest Mini communicating with your phone).
The only two workarounds I know of for this are:
- Use a phone or tablet as a Wifi hotspot. However, this will then use your mobile data plan, which could be expensive for you depending on the data plan you have with your carrier. Note you will need 2 devices to set this up.
- Turn on the Wifi hotspot feature on a phone or tablet. If you're not sure how to do this, refer to the manufacturer's support website.
- Connect a second phone or tablet to that Wifi hotspot.
- You can then use that second phone or tablet to setup your Google Home and connect it to the Wifi hotspot on the first phone or tablet.
- Invest in a cheap travel/portable router. If you have the option to use a wired connection, instead of a Wifi connection, connect this router to the wired socket. This lets you broadcast your own Wifi setup, that you can connect your devices to instead.
