Automator: Take screenshot > Prompt rename > Add date and time to name
Renaming and moving files with other actions in Automator usually involves saving the list of file items to restore later, then adding the other actions using the Ignore Input option as needed to keep from mixing up the various results.

If you are using a destination folder for the final result, you can use a workflow something like:


Folder Action receives files and folders added to { targetFolder }
Move Finder Items { To: destinationFolder } (to keep from triggering
the folder action again)
Set Value of Variable { Variable: fileItems }
Ask For Text (Ignore Input) (This will be text for the new name)
Set Value of Variable { Variable: newName }
Get Value of Variable { Variable: fileItems } (Ignore Input)
Rename Finder Items { Replace Text, Find: (the text "Screen Shot")
in basename only, Replace: (drag newName variable) }