smeeps If I try in privacy focused browsers such as Fennec, or Mull, the popup never opens and the authentication just fails.
That might be an issue with those browsers. Have you checked their respective issue trackers to see if others are reporting the same issue?
smeeps Same for if an app opens a webview to authenticate, the hardware key doesn't work in this webview.
I think you might be thinking of custom tabs, but in any case, try first to enable JIT in Vanadium (assuming Vanadium is your default browser) for those websites. For instance, for Tuta you can go to https://app.tuta.com and tap on the "site settings" icon in the left corner of the address bar, allow JavaScript JIT and refresh the page. Then go back to the Tuta app and try to sign in again.
If that doesn't work, I might be able to test those apps if you name them.