Whether or not Britain could have developed the atomic bomb (and yes, while they did initiate the first real nuclear weapon development program in the Tube Alloys program, later transferred to the American Manhattan Project along with the predominately German, Austrian, and Swedish refugee physicists and chemists who worked on the program), as pointed out by Polycarp, Britain was already hemorrhaging money to maintain the Empire even before the expenses of WWII. This, and various growing independence movements on the Subcontinent and Africa spelled then end of Britain’s globe-spanning influence.
One interesting but entirely hypothetical line of inquiry is what would have happened in Eastern Europe and Central Asia had Britain not been so preoccupied with packing up the bus and getting out of the empire business. Churchill was an early but constant voice about the danger of Soviet expansionism into post-WWII Europe, and his prognostication turned out to be largely correct. Unfortunately, the British public, fed up with over five years of warfare, had no taste for fighting another European land war with an enemy that had a number of strategic and numerical advantages, and the Soviet Union gorged itself at the buffet of Eastern and Central Europe with the tacit approval of the other Allies, much to the dismay of Poles, Czechs, Hungarians, the Balkans, East Germans, and so forth. Ironically, allowing the Soviets to absorb these nations prolonged its existence by decades as it drained them for labor, resources, and manufactured goods that Russia and the Ukraine itself couldn’t produce, as it was failing economically even before the costly involvement in the Great Patriotic War.
Stranger