Probably not yet, to be honest. But ten London boroughs have taken the plunge and are attempting to interact online with their residents, with varying degrees of success. Some of them are even occasionally interesting, or indeed useful. None of the ten are tweeting prolifically, so don't worry that your council tax is being wasted. Here's a quick summary of how they're doing.
Barnet: twitter.com/BarnetCouncil - 358 followers, 296 updates (first tweet May 2008) » Example: "Could Barnet be more creative in park flower design: Some time ago I made a photo survey of parks in Barnet, Bre.. http://tinyurl.com/dezm74" » Summary: Mostly a link to council news, but with the occasional human interruption » Also - Facebook, Flickr, YouTube
Camden: twitter.com/camdentalking - 178 followers, 193 updates (first tweet February 2009) » Example - "Power cut in the Hampstead Road area, some residents are without electricity: http://bit.ly/qVSL #camdencouncil #eustonroad #camden" » Summary - Started out as a snow update, and still regularly informing residents about changes to services » Also - Flickr
Haringey: twitter.com/LBHaringey - 70 followers, 15 updates (first tweet last week) » Example: "From News Feed: Extra refuse services available for Passover: Special skips will be appearing in .. http://tinyurl.com/d2na7v" » Summary: Very recent; mostly regurgitated newsfeed
Hillingdon: twitter.com/Hillingdon - 215 followers, 207 updates (first tweet June 2008) » Example: "Say hello to @CllrSSD - first Hillingdon councillor to use Twitter" » Summary: Announcements and chat, making a special effort for Comic Relief » Also - Facebook, Flickr, YouTube
Lambeth: twitter.com/lambeth_council - 99 followers, 31 updates (first tweet March 2009) » Example: "New initiative launches in Lambeth to stop knives getting into hands of young people http://bitly.com/hKrIY" » Summary: Still rather new, but almost all press release URLs
Redbridge: twitter.com/LB_Redbridge - 72 followers, 9 updates (first tweet February 2009) » Example: "Redbridge i, the council's innovative website wins a prestigious award http://tinyurl.com/d4faxc" » Summary: Fledgling service, intermittently updated
Southwark: twitter.com/lb_southwark - 180 followers, 52 updates (first tweet December 2008) » Example:@loobs0 "We'd laid down 500 tonnes of grit between Sunday afternoon and yesterday afternoon. Had another 60 people out gritting last night..." » Summary: Mostly updates on council events and services » Also - Facebook, Flickr, YouTube
Wandsworth: twitter.com/wandbc - 209 followers, 84 updates (first tweet December 2008) » Example: "Dry cleaners go green: Two-thirds of Wandsworth's dry cleaning operators attended a council training workshop on.. http://tinyurl.com/79vxkt..." » Summary: Pure press release sausage factory
If you live elsewhere in the UK, or if you want to discover whether your local councillor is tweeting, check out the ever-growing directory at Cllr Tweeps. It's future of local democracy, you know (or perhaps a pointless micro-audience dead end).