This information is correct for the 2008 Show. There may be minor changes for 2009 - we won't know until a day or two beforehand - but in general the affected area stays the same and the timings only change a little bit.
Most roads in and around the City of London are going to be completely closed for most of the day of the Show, admitting no traffic whatsoever. Most parking bays close to the Route will be suspended from 6.30 in the evening before.
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In short, every road east of the Kingsway, south of Holborn and the Barbican, west of Bishopsgate and north of the river will be closed for most of the day of the Show. Most of the City proper is accessible again after 4pm, but the area between Farringdon Road and Kingsway remains closed until 6pm. The same goes for Blackfriars Bridge and the stretch of Upper Thames street that heads east from there towards Southwark Bridge, both of which are closed until 6 for the fireworks display.
London, Southwark and Waterloo Bridges will be open as normal, but access to the north end of Southwark Bridge will only be from the east, along Upper Thames St: all other approaches are closed.
There will also be traffic restrictions on many roads in the region extending north to Old Street and Clerkenwell Road, and you can expect heavier than usual traffic in the surrounding area as cars are rerouted.
Full details of the affected areas can be found on yet another handy downloadable map.