Here's an attempt at an interesting article on the robustness of UK telecoms:
http://www.theregister.co.uk/2008/11/28/wireless_survivors/

To summarise: mobile phones are not intended to be robust, but landlines are and ought to last 14 days without grid power or human attention. Their use might be restricted, though. The government has additional communications resources, and could use radio to inform the public.

The article doesn't say how long grid power itself could be expected to last. There is some valuable criticism in the comments section.

The context is a TV drama called "Survivors", about disease killing off 99.9% of the population. That production has virtually nothing of value to this site, in my opinion (and I think it's rubbish artistically, too).