The sun has not yet set on the British Empire, and is unlikely to for the near future.

This question was extensively covered by xkcd.

Britain [still owns] 14 overseas territories, [which are] direct descendants of the empire.

These 14 territories are: Akrotiri and Dhekelia; Anguilla; Bermuda; British Antarctic Territory; British Indian Ocean Territory; British Virgin Islands; Cayman Islands; Falkland Islands; Gibraltar; Montserrat; Pitcairn Islands; Saint Helena, Ascension and Tristan da Cunha; South Georgia and the South Sandwich Islands; Turks and Caicos Islands.

The sun has never simultaneously set on all 14 territories at the same time.

Again, from xkcd:

Every night, around midnight GMT, the Sun sets on the Cayman Islands, and doesn't rise over the British Indian Ocean Territory until after 1:00 AM. For that hour, the [...] Pitcairn Islands in the South Pacific are the only British territory in the Sun.

However in roughly 400 years on April 30th 2432, the Pictairn Islands will experience a total solar eclipse. Still, during this time the sun will still be shining in the Cayman Islands and the UK, so the sun will still be shining over the British Empire.

So unless the British Empire loses more territory or the current borders change, the sun will continue to shine over the British empire for millennia to come.

It is worth noting however that the Pictairn Islands has a total population of about 50 people and in 2004 1/3 of the adult male population was convicted of child sexual abuse. So due to population problems or some other disaster they may become uninhabited forcing an early sunset on the empire.