What do you mean by "rule the world"? If you mean "physically conquer most of the world," then lol. France and Germany would balance against it, and deny Britain land control of Continental Europe, the most economically important part of the world until the late 20th century.
Re empire building, the problem with drawing wanked maps of British colonialism everywhere is that, as noted by others, it wasn't terribly useful. If you believe the excellent but problematic Pseudoerasmus, Britain didn't care much about African colonies (
link), and readily compromised with other European colonial powers about divvying up Africa, provided its interests in India were not challenged. The biggest pressure for settlement expansion did not come from the center, just as the westward expansion of English settlement in the US came from the American colonists, especially the backcountry ones, and not from Britain. Cecil Rhodes defined himself in opposition to centralized rule from London, and championed white settlers in Africa against the metropole, which he viewed as meddlesome. Few to no extra settlers would be available for such expansion in the ATL, which means there's no reason to expect deeper white settlement in Africa than in OTL, and hence no reason to expect drastically more UK colonialism there.
One thing that would change from OTL is that the transition from British to American political dominance would be smooth, and under the aegis of a single state. In OTL, this transition involved two world wars and an interregnum without a dominant world power from 1914 to 1945. Of course, the US had a much larger economy than Britain even in 1913, by a factor of more than 2, but it had very low military spending, so it wasn't as globally powerful as Britain was. In the ATL, Germany would never challenge the UK on its own, but instead gradually form something like a Britain-less EU to balance against it.