How careful should you be going to London

Greater London is served by three police forces; the Metropolitan Police which is responsible for policing the vast majority of the capital, the City of London Police which is responsible for The Square Mile of the The city of London and the British Transport Police which police the rail network and London Underground.
Overall crime is down in London, but you have to look at the individual areas that make up the Metropolitan Area. For example, the Towers Hamlet rate is up while Southwark, right next door is down. You can find specific crime rates for the area you are visiting, at this site http://www.met.police.uk/crimefigures/

The increases in robbery were largely attributed to the rise in youth on youth robberies across London with particular focus around schools and transport  interchanges and increased usage and ownership of items such as mobile phones, one of the most commonly stolen items .

I have gone to London many times and never had a problem. I find London a pretty safe city but you always should be careful no matter where you go.