I have just loaded the game into a new laptop which has a CD-Rom. However when I go to run the game it is asking for a CD, which as a digital download does not exist. How do I get it to run from the HDD?

Actually, GoG's digital download of X-Wing and TIE Fighter contain a CD-image and a cue-sheet. They're called "game.gog" and "game.ins" respectively. You can try to burn the image on a CD but you'd have to alter the extensions.

X-Wing vs TIE Fighter and X-Wing Alliance don't have any CD-images.

You should write which game this is. But it would indicate some problem with one of the patches used to make the game run without CD.
I read some reports about this before, but I don't know about any definite fix.
Not sure if GoG support can maybe help.