In 1956, Britain did try to fight for her empire. After Egyptian President Nasser took over the Suez Canal (one of Britain's largest overseas assets), the British, the French, and the Israelis attacked them in an effort to get it out of Egyptian hands. This action was universally vilified, and their actions were condemned both by the US and by the USSR in the UN.

Simply put, the British did 'fight' for their empire because they were no longer capable of acting alone. The United States' support was essential for projecting power around the world, as was shown in the Falklands War, where the US provided material and logistical support.

If you want to read about Suez, the so-called 'Empire's Last Stand', I'd recommend the book "Suez" by Keith Kyle.

Coming out of the Second World War you could argue that Britain had already ceased to be a global superpower. Infrastructure including transport links, as well as factories, and housing, had been significantly damaged due to German bombing across the country. The sheer cost of rebuilding all of this in the decades after the war meant that Britain simply could not afford to keep the vast majority of it's colonies.