Thank you for your help and you might be iterested to know that I have solved my problem.
In the Belkin router set-up utility there's a function called 'MAC Address Filtering' about which I was oblivious until in desperation one day I looked it up on a PC Dictionary site. (There was nothing in the Belkin help file) It's all about Media Access Control which is another American acronim meaning internet access. Well, mine was disabled by default which was of course preventing the Laptop suiccessfully pinging the router. It would let me ping the two other PC's on the network but not the internet. I just don't know how it was set-up and worked initially. I disabled it and up it came as good as gold.
What amazes me is that I made four very long calls to Belkin help who never came close to the subject, not to mention several other Forum posts. I've been on and off this problem for about six weeks now on the assumption that one day it will hit me in the face!! One good thing came out of it....I've learned quite a lot about networking.

I havn't spotted any other Posts on Daniweb about similar problems but I hope my experience will be a guide and perhaps a solution to others.

PS Have you seen my post on Microsoft Fax in Windows Software?
Can you help please?

It doesn't surprise me that Belkin couldn't help you. I had a similar experience with them and I will never buy another Belkin network product. Only in my case I'm sure it was a deliberate act to leave certain information out of their product documentation and from their website. The tech knew instantly what the problem was, and my recourse was to return the product to Belkin and wait for a refund, of course that didn't get me back all the hours I spent fooling with it. The reason it was left off the package was simple, almost no one would buy the product if they knew in advance.