I have successfully upgraded to Windows 10 one week ago, no problem with Elements 13, Lightroom or Photoshop CC.
So yesterday, I decided to reinstall my Elements 12 version: no problem in the installation and upgrade to 12.1.
Will that work for you? There are still problems for Elements 13 as you can see in another discusson on this forum; I don't believe the Elements version is relevant in those problems.
To transfer your activation, you have to use the Help menu, 'Sign out' on the old compuer, and you have to 'sign in' on the new one with your registration code.