At its height in the early Twentieth Century, the British Empire spanned a fourth of the world. That is why it was said that the sun never set on the Empire: it was always sunny in at least one part of Britain's possessions.
Except for a few Overseas Territories--mostly small islands--Britain has departed from its empire. Can it still be said that the sun never sets on it? According to Randall Munroe of xkcd, yes.
The biggest daylight coverage is provided by the Pitcairn Islands in the south Pacific Ocean. Those islands will experience a solar eclipse in April of 2342. Will the sun finally set on Britain on that date, assuming that it retains its current territory? No:
Luckily for the Empire, the eclipse happens at a time when the Sun is over the Cayman Islands in the Caribbean. Those areas won't see a total eclipse; the Sun will even still be shining in London.
In fact, no total eclipse for the next thousand years will pass over the Pitcairn Islands at the right time of day to end the streak. If the UK keeps its current territories and borders, it can stretch out the daylight for a long, long time.
But not forever. Eventually—many millennia in the future—an eclipse will come for the island, and the Sun will finally set on the British Empire.
(Images: Probert Encyclopedia, Randall Munroe)