Whenever we need to restart our Thomson TG585 router for any reason (either via a Telnet/Web command, or via power off/on), it automatically reconnects to the internet after the restart. Recently though, we suffered a broadband fault on our line for a day or two, and after the line was fixed the router would no longer automatically reconnect to the internet when powered off/on.
This was quite a puzzle, since there's no obvious configuration setting that can be turned on and off ("Connect to internet on startup" or suchlike). I couldn't find any answers online, but what I found by experimentation was that the router seems to silently save an implicit setting **when the restart function is used**, and **via the web interface only**, which records the current internet connection state and restores that state after subsequent restarts (of any kind, including power-off). And while the broadband fault was being investigated, our ISP had us restart the router via the web interface, while the internet connection was (obviously) down.
So the fix, in summary: ensure the router is connected to the internet, and then use the restart link **in the web interface**. After this one-time action, our router now always automatically reconnects to the internet after any restart, whether via the web interface, telnet or power-off.
Hope this helps someone else with the same problem.