It doesn't really matter to our dimension the British Empire never existed in another reality. At least it doesn't until somebody from one of those timelines comes to visit. This week we explore portals to other universes.
Like most of the major Alternate History tropes this one is actually quite old. In fact, the Sidewise Award for Alternate History is named after Sidewise in Time, a 1934 story by Murray Leinster where the planet is suddenly and disastrously made aware of alternate universes when parts of the earth are replaced with areas from other timelines. Legionnaires appear from a world where Rome never fell and longboats show up in coastal cities from a world where the vikings colonized North America.
While Leinster's story had a lasting effect on the genre, H Beam Piper's Paratime series is probably closer to the version of this trope that we're all familiar with. In Piper's stories, beginning with 1948's Police Operation, a powerful organization from an advanced society has the ability to travel between alternate universes and uses it to collect resources which have become scarce in their timeline.
The Paratime organization is also the inspiration for Harry Turtledove's adolescent Crosstown Traffic series. The titular Crosstown Traffic company is also an organization dedicated to collecting resources from alternate universes, but Turtledove's series tends to concentrate more on the alternate universes than the home timeline of the Crosstown Traffic agents. Usually the protagonists are a pair - one from the advanced timeline and one from the alternate universe that novel is set in. Turtledove has used this formula to explore worlds where Imperial Germany won WWI and rules the world or where Russia won the Cold War.
Not all alternate universe stories have such a large scope. Sometimes traveling to another timeline is a one time event. In Brad Ferguson's The World Next Door two worlds, one ravaged by Nuclear War and another that's about to be, come into contact with one another. S.M Stirling's Conquistador deals with an accidental incursion between worlds that results in a 1940's army brigade setting up a new Californian empire in a world where Europeans have not yet discovered America.
Philip K. Dick even gets in on the trope with The Crack in Space, in which an overpopulated future society tries to get rid of their excess citizens by colonizing an alternate universe filled only with primitive tribes of Homo Erectus. At least they were primitive until one of the humans starts teaching them more advanced technology.
More recently Charlie Stross's Merchant Princes series, beginning with The Family Trade follows a group of humans able to jump from world to world using their power to create a powerful (and morally questionable) trade organization across universes. The series was initially billed as fantasy (since the major timeline of the first novel stalled out in the medieval period) but the series is really science fiction that concentrates on the economic aspects of trade between worlds.