Chromecast, Google Home and Google Nest devices don't support these kind of Wifi systems, known as captive portals, that require you to login first to be able to access the internet.
There are only really 2 possible solutions for this:
- Check with whoever manages your Wifi if they have a process to whitelist your device. They do this by allowing the unique MAC address of the device. Not all providers will be setup do this however. In addition, even this may not work on some Wifi networks, as the the administrator may have enabled a security feature called client isolation, which prevents different devices from communicating with each other (i.e. your phone communicating with your device).
- You mentioned you may not be able to do this but I'll explain it anyway. Although unsupported by Google, you can try to 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.
If you have the option to use a wired (ethernet) connection to the network, instead of Wi-Fi, you could invest in your own router and connect your devices to that instead.
