US and British deaths in the Afghan War were low because it wasn't really a US or British war. The US and British provided air cover for forces of the Northern Alliance who fought long and bloody battles over a period of two months to capture the major cities.
The first major ground battle started in November 9th 2001 when the Northern Alliance started their seige of Mazar-I-Sharif. When the Taliban tried to take Mazar in 1997, 2,000 Taliban fighters who were captured were later sumarilly executed. In revenge when the Taliban finally took the city in 1998, they engaged in what Human Rights Watch has described as "the single worst examples of killings of civilians in Afghanistan’s twenty-year war", with thousands of ethnic Hazaras rounded up and summarilly executed. While its not known how many were killed in the US backed siege, reports from soon after the Northern Alliance entered the city showed a level of factional killings that resembled the previous times that the city fell. RAWA has ducumented the mass slaughter of Pakistani nationals who were trapped in the city. 10 UNICEF trucks were also captured by the Northern Alliance and some of the drivers were executed.
The seige of Kabul started almost immediately after the fall of Mazar and by November 12th, Kabul had fallen to the Northern Alliance. The corporate media showed glowing pictures of Afghan men now free to shave their beards and Afghan women no longer required to wear burkas. Many Afghans worried about the past human rights abuses of the Northern Alliance forces which now controlled the city. Farooq Tariq of the Pakistan Labour Party wrote that "in order to force the Taliban out of Kabul, Washington had to rely on supporting one group of Afghan religious fundamentalists against another. The NA might make some changes in its outlook initially, but it will not change its fundamental aim of maintaining an Islamic state in Afghanistan."