OpenType/Unicode fonts were not used in Word and/or in InDesign. Check your Lao fonts; are they Unicode versions? (Unicode is the standard character set used worldwide. It allows us to retain the characters as the text moves from one program to another, from one browser to another, etc.)
Don't copy/paste into InDesign because you'll lose some of the behind-the-scenes coding of the Lao glyphs. Instead, use InDesign's File / Place command and check the box for Import Options to control how the text is imported into your layout.