Even though Firefox does not support U2F by default, there's a working U2F plugin that works well (tested with Github), but Bitbucket won't even allow the use of U2F if it does not see Google Chrome, and prints this message instead:

"Your browser doesn’t support the FIDO U2F standard yet. Get the latest version of Google Chrome if you want to use a security key device"

It's a shame this is based merely on User Agent and not the actual capability of browser, which can obviously be extended with a plugin.