The two times I have been to London and the UK, I have felt very safe and had no problems with pickpockets. Never the less, as a traveler you should take proactive steps to protect your self. Read over my safe travel for Spain page which covers information on pickpockets.
The US Department of State Consular Report on Ireland - Provides information on traveling to the Republic of Ireland and current safety issues.
The US Department of State Consular Report on the United Kingdom - Provides information on traveling to Northern Ireland and current safety issues.
I do strongly recommend (when not immediately needed) one keep their passport, most of their money and back up credit card in a money belt under their clothing. Make copies of your documentation and keep them separate from the originals. You will need to have your pastport ready at the airport, country entry and to exchange money. You might also find you need to provide an ID to use your credit card.
If your hotel provides a safe, consider using it.