The BBC does use the Met Office for its weather information. (
http://www.bbc.co.uk/weather/metoffice.shtml)
However, the two things no one here has considered are the location and the time - just where is London, and what time is the forecast for? Also, at what time was the forecast generated?
On the BBC site you can enter the postcode you are interested in and will get different numbers (even though they may all be "in London"). Also the forecast time and type may differ - one may be a midday report, a mid afternoon report, worst weather report, average conditions report. lastly reports are constantly updated, so you also need to check the time at which the feed used was generated.