IANAL, but in most legislations you do not have to register a trademark for it to be valid. It is important that you defend you trademark, and you will lose your right to a trademark if you fail to do so.
I do agree with those saying that in the wider audience, GNU screen does not mean anything, including (probably) the large armies of Windows developers. Besides that you should use tmux anyway ;).
the issue is that you can't easily trademark a common word. It's possible (e.g. Apple for computers) but you run into problems. Apple ran into some notorious trademark problems.
Trademarks don't need to be officially filed to be real in common law -- you just have to use a distinguishing mark for a certain period of time. so the GNU/FSF screen would have rights to Screen in the computer program industry even if they haven't officially filed. Filing just helps you defend a trademark, it's not strictly necessary.
