In reply to Solsbury:
Sounds like you do not have a backup of the photos. If they are on the external hdd then the files will be clearly visible.
If backed up by windows, then there is (v7 anyway) an option in the back up program to restore from backup. That would mean just connect the external hdd and go to the backup program and click restore files. However, I am bit lost with what you are saying, so I wouldn't just jump to that until further things are established first as further clarification is needed of what you are trying to achieve.
What do you mean by reinstate - do you mean restore photos to internal hdd from external? If so why? Where are you trying to retore to and why? (I read it that you don't have the new PC yet). Do you not still have the originals on the PC? If so are you just testing the integrity of the backup before you go to the new PC?
If you are trying to view the files stored on the external hdd via my computer then from what you have said its sounds like you do not have photos backed up but the Lightroom catalogue of them. If that's a possibility then before any attempt to restore (no need to restore if just testing the integrity of back up or you could lose the originals), check the contents of the external hdd via my computer. Lightroom does not store the photos themselves, but the link to them and the catalogue of them. If you are saying no jpegs files are visible on the disc, then they have not been backed up and I wouldn't do a restore at all.
If the photos are on the internal hdd, but not the external one, then it's a case of starting again with a backup. If all you are doing is a photo backup, then copy and paste to the external hdd maybe be the simplest. If your wanting the Lightroom cat then you would have to copy that across as well.
Sorry if I've misunderstood what you are saying.