The computer that works has an IP that matches the router IP, which is also shown in System preferences: 88.157.222.106 and 88.157.222.254 respectively. The computer not working has another IP altogether. Where would this have come from? The DHCP cannot be working properly. It does not show a router IP.
It is strange, as it was working OK on another home connection, all the time on automatic, using DHCP, and worked at my home fine before that.
Manually setting an IP won't work, as the wireless base is giving out IP via DHCP. I want to keep it automatic for convenience, and in any case, as you say, there is clearly something wrong that needs to be fixed.