The Soviet Union had begun their nuclear program even before the end of World War II and accelerated it dramatically, with nearly unlimited resources assigned to it and maximum priority, after Hiroshima. They were desperately afraid of the United States having a major advantage over them. They detonated their first bomb in 1949, four years after they made the program a top priority.

The United Kingdom, on the other hand, did not start their own independent nuclear program until 1947, when their participation in the American program ended, and opted to do it on the cheap. They did not feel they had the resources to make a top priority, and their approach was relatively conservative. Still, they detonated their first bomb in 1952, which is to say, five years after they started off on the nuclear path, which isn't so different than the Soviet case except that they waited longer before starting it.

France is a separate case, wrapped up in their view of what it meant to be a relevant power in the Cold War, and their own wavering feelings about European alliances (e.g. NATO) and desire to maintain a certain independence. They did not decide to go nuclear until 1954. They did not pursue and all-out program — it doesn't seem to have been motivated by an urgent sense of existential dread, in the way the US and Soviet programs were, and they did not detonate a weapon until 1960, which is to say, six years after beginning their program.

Which is only to say, for nuclear weapons programs there is a direct relationship to how much priority and funding is given to them. The wartime Manhattan Project, done with unlimited funding, top national wartime priority, and motivated initially by a fear of Nazi nuclear weapons, is still the world's fastest nuclear program (it started in late 1942 and had usable weapons by the summer of 1945, which is to say, about 2.5 years). The Soviet program, motivated by similar fears and with judicious application of human (gulag) labor, with some help from espionage sources, managed to do it in 4 years, still a very "good" time. The British and French programs were neither so lavishly funded nor so urgently pursued, and so they had "cheaper" bombs but the amount of time to get them increased. Just for reference, the Chinese started their program in 1955, and had a first weapon in 1964 — nine years, in part because there was some chaos in their early program (along with taking place during the upheavals of the Cultural Revolution, the Sino-Soviet split caused them a lot of early grief, and the fact that they were much poorer at the time than any of the other nations that had developed weapons so far).

As nuclear programs are large and complex "technopolitical" entities (in the jargon of the field Science and Technology Studies, which is meant to emphasize that the technical, organizational, and political components cannot be segregated or separated when analyzing these things) that unfold over many years in complex circumstances, there are no obvious and quick answers to "why was country X in time A able to do something that took country Y in time B more time to do?" The answers I've given above are very superficial in this respect, and are just meant to give a flavor of the different circumstances.

A nice overview of various world nuclear programs, successful and not, is Jeffrey Richelson's Spying on the Bomb. It should be noted that there are several scholars in the field of International Relations who debate and disagree on what factors lead to rapid, slow, successful, unsuccessful, etc., proliferation attempts; e.g., Jaques Hymans argues that the organization of the nuclear programs matters a huge amount, while, say, Alex Montgomery points out that while it might seem that getting external "help" (e.g. the Soviets helping the Chinese) would speed things up, it seems to actually slow things down (because being dependent on technology you did not yourself create means you cannot adapt to difficulties very quickly), among other arguments.