so that suggests the wireless layer is OK in that you can chat to the router wirelessly and see its web page.
When it's "off" you could try doing 'nslookup www.bbc.co.uk'
from a command line to see if that works, and if it does try 'ping www.demon.net'
to see if you can send packets across the internet.
If 'ping 192.168.whatever' on a command line works to the router address, and 'ping 188.8.131.52' doesn't, then your router has issues with routing. If these both work but 'ping www.demon.net'
doesn't then it's a DNS issue.
MaxDSL - goes as fast as it can and doesn't read the line checker first.