How does the communication between the USB key and Google in a browser work? Will it work on all operating systems and browsers?
As for the operating systems: in order to use it on your phone, you'll need to be careful to use a YubiKey with NFC support and have a phone that supports NFC. You won't have any trouble using it on Windows/Ubuntu/macOS. As for the browsers: Chrome/Chromium works fine. Firefox has an addon that adds security key support[0]. Unfortunately, this addon has a bug which causes high CPU load[1]. Firefox is also working on…
WebAuthentication will be supported in all major modern browsers [1], it just takes some time to implement.
[0]: https://bugzilla.mozilla.org/show_bug.cgi?id=1065729#c254