If it works with previous versions of Windows 10, it will undoubtedly work with the latest Windows 10 update.
But I can't promise Premiere Elements 10 will work on Windows 10. Version 10, after all, was released in 2011, and a lot has happened since then -- both in terms of operating systems and video formats. Though you should be able to at least run it in Windows XP compatibility mode.
Though I would very much recommend an upgrade to at least version 12. The program is more stable, more versatile. easier to use and can take better advantage of 64-bit operating systems. It may not be worth struggling to get version 10 running -- a version of the program that's not even capable of working with a number of current video formats (phone, GoPro, etc.).