I'm not just asking about the making of the atom and hydrogen bombs — although this book would describe their origins. I'd also be interested in learning about the development of nuclear weapons in the USSR, the use of nuclear missiles via submarines, the history of ICBMs, and maybe even nuclear disarmament. I'd also love to learn about the history of America's policy with regard to the use of nuclear weapons. When was it decided whether a response would be proportionate or unrelenting? What is the procedure for carrying out a nuclear strike?
If there isn't one book that covers all of these, are there a few books that cover these topics?
The problem with any book that tries to be comprehensive is that you inevitably must leave things out. You must focus here or focus there, and sometimes the more you try to do, the less you accomplish. So Gerard DeGroot's The Bomb: A Life, is sort of an attempt to write an entire history of nuclear weapons. And I have to say, I don't think it quite pulls it off. It ends up treating some things in inordinate detail relative to their weight in the overall history (do we really need a huge account of Heisenberg's work in WWII, given that he didn't produce a bomb?), and other things it just breezes over too quickly because nobody (probably) wants to read a 3,000 page tome.
Hey, you say, I might want to read a 3,000 page tome! Chuck Hansen's Swords of Armageddon is a 3,068 tome that tries to be the be-all and end-all of nuclear weapons... but is only really about the United States. That's not a flaw, per se, because that's his goal, and there's no way you could write as comprehensive a work on the Soviet Union, China, UK, France, India, Pakistan, Israel, North Korea... the US has declassified the most information about its nuclear program, by far, than any other state. But this just goes to illustrate how hard you'd have trying to write a real book that covered the entire history of nuclear weapons well — even if you didn't limit yourself by space, you'd still necessarily have omissions and areas of focus. And even Hansen's book doesn't really cover all of that material equally well — he's got almost nothing on the broader politics and strategy, for example, because he was mostly interested in the technical details about the warheads and the delivery systems. And he wasn't really trying to write a readable narrative, so much as a collection of a huge amount of research.
So not only does the book you're looking for not really exist, it probably couldn't exist. That being said, there are many fine books that cover different aspects of nuclear history well, and some that cover a lot of ground well. I am fond of recommending Schlosser's Command and Control, because even though it is in principle focused on one aspect of nuclear history — problems relating to authority and accidents — that ends up being a way to get at a lot of different kinds of stories pretty well, and covers quite a gamut. It is mostly focused on the US history, but it does dip a toe into other histories as well.
If you are looking for a long monograph on the evolution of nuclear strategy, Lawrence Freedman's The Evolution of Nuclear Strategy is, as the title suggests, just the ticket.
If you are interested in nuclear culture and attitudes, Spencer Weart's Nuclear Fear: A History of Images, is excellent. If you're interested in nuclear secrecy in the US, there's a book written by some guy on that subject too, I've heard, with some kind of silly title like Restricted Data: A History of Nuclear Secrecy in the United States (but it does have a cool cover, I have to say). There are books on the Soviet nuclear program (e.g., Holloway's Stalin and the bomb), the Chinese nuclear program, the Israeli, Indian, and Pakistani programs (what we know about them, anyway), and so on and so on, even OK books on the North Korean program (I found Ankit Panda's Kim Jong Un and the Bomb very informative).
Want to know about the US game theorists and war planners? Fred Kaplan's Wizards of Armageddon is a classic. Want to know about the history of the nuclear winter hypothesis? Lawrence Badash's A Nuclear Winter's Tale.
Which is just to say — there's a huge literature out there. When someone asks me what they should read as a beginner, I tend to point them to Rhodes (The Making of the Atomic Bomb), Schlosser (Command and Control), and Weart (Nuclear Fear) as a start. But that's just a start, and I've picked those because they cover a lot of ground between them, and are very accessible. To go beyond that requires making choices about what you care about and what you don't.
I highly, HIGHLY recommended “Command and Control” by Eric Schlosser. The book’s main focus is on nuclear weapon safety and accidents involving nuclear weapons, but there’s a ton of fascinating descriptions in the book on everything from how the launch keys work to enemy missile detection and everything in between. If I recall correctly there was lots of material on policy. One of the most interesting books I’ve ever read!
While it's not a comprehensive history of all nuclear weapons programs, you may enjoy "Target America: The Soviet Union and the Strategic Arms Race, 1945-1964" by Steven Zaloga, which offers a broad overview of the Soviet Union's nuclear program, including the development of nuclear capable ICBMs, SSBMs, and nuclear capable bombers.