Disable the option Display full paths in project list at Configuration - Editor Display - Miscellaneous.
Some project settings (relative path to ...) also disappeared from the project settings dialog. But they are still recognized, if you specify it directly in the project file .prj. When UltraEdit is already started and the project is not loaded, open the *.prj file with the file open dialog and specify the "old" settings according to your needs:
Use Relative Directory=1
Relative to Project File=0
Include Sub Directories=1
And you have to edit the paths of the files already present in the project file. Then save and close the project file and open the project. Now the project works again with relative paths - relative to the specified project directory or the project file. And in the File Tree View the project files are then also displayed only with the file name.
PS: Opening an UltraEdit *.prj file via File - Open for editing might not be possible on Windows 8 / 7 / Vista depending on version of UE/UES. Open the project, open File Tree View (UE) or Workspace Manager (UES), select the tab Project, right click on bold project name at top and left click in context menu on Edit Project File.