The British Empire was the name used by the United Kingdom during its time as a colonial power.
Yvonne Hartman of Torchwood One hoped to use alien artefacts to rebuild the British Empire, much to the Tenth Doctor's confusion. This plan was abandoned after the Torchwood Institute's "old regime" was destroyed in the Battle of Canary Wharf in 2007. (TV: Army of Ghosts) Jack Harkness, who had worked with Torchwood for over a century, reshaped the remnants of Torchwood in the Doctor's honour. (TV: The Sound of Drums)
However, the British or English Empire monicker was also used to denominate several alternate variants of the United Kingdom. For the most part, these were usually more extreme versions of the British government, totalitarian states subject to xenophobia. One of these parallels was seen exploiting and torturing a crippled version of the Sixth Doctor (AUDIO: Jubilee) and another one was briefly created by Sabbath in which the Empire was murdering key personnel essential to the development of the computer. (PROSE: The Domino Effect) These timelines, for obvious reasons, were seen by the Doctor as mockeries and were destroyed or the events leading to their creation undone.