Thanks ORAC, but I got there just now through a different route. And I'm not sure that the Hosts solution would work where IP addresses are dished out by the DHCP Server which is my ADSL Router.
The step I was missing out was Mapping the Drives of the other computers to a drive letter on the "Server" where I am backing up the data. (Quotes on "Server" as this is a P2P LAN).
So I'm off and running now and will just need to restore the mapped drives every time the Network is switched on.
Mapping drives is a process I've never found the need of in the past. But once it is done, the "Client" drives appear as additional drives on the "Server" and can be dealt with as though they are physically there.
Thanks again for thinking about this for me.