Further update...
I had a bit of free time this afternoon so I moved the offending computer from its usual position to my computer workbench (aka the dining room table) and started to troubleshoot the problem and if necessary, re-install Windows.
After re-seating the RAM I removed all unnecessary PCI cards (I've learned from bitter experience that the fewer of these the smoother the opsys installation) and for good measure changed the graphics card. After initially booting to a BIOS screen that informed me that my CPU frequency settings had changed, I played around with the BIOS settings but basically ended up with it as it was before the trouble, restarted and it then booted normally. Then shut down again and reinstalled the network card and guess what... it recognised my Talktalk network.
I had a bit of a problem with the 'Windows cannot find a certificate to log you on...' caption but then I remembered to uncheck the 'Enable IEEE802.1x authentication' box in Properties and I am now on line as if nothing had happened but sadly, no wiser as to the cause of the problem.
As I mentioned in my original post, this machine is not used very much and when I checked the Firefox version I saw that it was running Firefox 2!
Thanks again for all the suggestions.