You can try going to ProgramData\Adobe\Lightroom and see if there is a lrreg file for Lightroom 5. If there is, you can open that file with Notepad and retrieve your serial number. The ProgramData folder is a hidden folder, so you might have to change your folder options in order to find it. If that doesn't work then you can go to your Adobe account online and your information should be available there.
Many thanks for the suggestions Jim.
Unfortunately I cant find an lrreg file anywhere, or even the ProgramData folder (I have selected to show hidden folders/files).
My laptop has two drives, the C drive (which I think is where the O/S installed) I guess was over-ridden when I had to re-install windows, although there is a folder called 'windows.old' which I presume is a back-up of the old o/s, though I have searched in there to no avail.
The actual copy of LR was originally installed onto the second (D) drive, which is the larger of my hard drives where I tend to install most of my aps and save my own personal files. I dont know (or really understand) how much of this would've been changed when I re-installed windows. I did go for the re-install option that said it would keep my personal files intact, and it does seem to me that a lot of my own files are still there on the D drive - but possibly not all of them. So I presume that the re-installation mainly effected files on my C drive. But i'm guessing.
Either way, there is a LR folder still on the D drive with a ton of files in it. Like I say, I was able to open the software without having to download or re-install anything further, and I was able to locate my orignial LR catalogs and load them up. So it looks like most of the LR stuff is still on my laptop even after re-installing windows. Everything, except for the serial number!!
I dont think I ever registered LR with my Adobe on-line account. It was a free copy of LR that came bundled with my printer and some other software (premier elements, if I recall) and I only installed it our of curiosity (I mainly use Photoshop CS5) and probably never expected to become so reliant on it!! But over time I started to use LR more and more, but just never initially registered the software with Adobe. You live and learn... looks like it could be an expensive lesson as I dont think I will be able to go without LR now that I have spent months (years!) cataloging, keywording, etc