There seems to be a confluence of issues here. The main problem is actually ReCAPTCHA. At least for me, it put me through multiple rounds of prompts to select images of different things. That's a sign that the algorithm isn't working. The sheer fact that it prompts to verify that you're a human at all, means that the algorithm isn't really working. The timeout error is actually complaining about the verifying the ReCAPTCHA response, so the whole issue here is ReCAPTCHA.
That said, it seems to only be affecting Windows 10, and even then, only some installations of Windows 10. The pop up login is actually an OS control, which should be using the default system browser, which in Windows is IE or Edge. I'm actually running the new Chromium version of Edge, which replaces the old Edge. My best guess is that it's likely an issue with Chromium Edge sandboxing the browser control, causing the ReCAPTCHA prompt to not be able to run its heuristics on the machine properly.
Long and short, if you're having issues with this, try it on a machine not running Windows 10, or if Windows 10, with out Chromium Edge. Unfortunately, if this is the issue, there's no way to workaround it, as you can't downgrade Chromium Edge back to old Edge, so if you've already moved over, you're stuck.