Define the British Empire.
It was never an actual 'thing'.
Do you mean British colonies abroad? Those still exist.
Do you mean Britain going around and talking about the British Empire? There you have to somehow stop our massive shift to the left in the 20th century. On this one I'm not sure of an actual date, I wonder does anyone have a clue, anti-oxford dictionary style: When was the last time a British government talked about the British Empire in a non-historic sense?
Do you mean the only actual empire part of it, the Indian Empire? There I think you'll need to do something to make Indian independence a bit more right wing dominated and have the royal family be more popular in India, then I don't see it as being too unfeasible you could have India officially keeping an emperor/press who also happens to be the British monarch.
Though the popular version of history likes to paint the end of British overseas colonies as Britan 'losing' them due to nationalist movements the truth tended to be quite the opposite, it was the British working class increasingly being able to utilise their vote that did it. This meant British politics moved sharply towards a more domestically focussed left and an idea of 'why spend money on poor people half the world away when we have poor people here?'