A group of UK researchers and geeks have put up Iraq Body Count, to keep track of the civilian deaths in Iraq:
This is a human security project to establish an independent and comprehensive public database of media-reported civilian deaths in Iraq resulting directly from military action by the USA and its allies in 2003. In the current occupation phase this database includes all deaths which the Occupying Authority has a binding responsibility to prevent under the Geneva Conventions and Hague Regulations. This includes civilian deaths resulting from the breakdown in law and order, and deaths due to inadequate health care or sanitation
The U.S. military has refused to provide such counts. Don't know the position of the British military.