First, the situation is not as grave as he makes it out to be. There are two types of nuclear threats. There are dirty bombs where nuclear material (i.e. plutonium, uranium, cesium) is blown up by conventional explosives like TNT, c-4, petn. There are also bombs with a nuclear chain reaction (nuclear bombs) where the nuclear material itself expands at trillions of miles per hour incinerating anything in its path. Dirty bombs are destructive; nuclear bombs are catastrophic and life-obliterating. Dirty bombs don't threaten the continuation of the human race, nuclear bombs do. Fortunately for us, Allison overestimates the ease of setting off a nuclear bomb. Based on my reading of the highly-regarded Richard Muller's "Physics and Technology for Future Presidents," you need a lot of highly enriched uranium or plutonium with highly advanced (nation-state level) explosives. Terrorist need to go through many steps to set off a nuclear weapon in the US or an allied country: getting a large quantity of nuclear material, transporting the nuclear material, getting high level explosives, transporting the high level explosives, assembling the weapon, transporting the weapon, setting off the weapon. It is a long process in which there is much time and opportunity for the good guys to stop them. All and all, we shouldn't deny the real danger, but we can also sleep at night and live in populated areas.
Second (mistake), when discussing how to prevent states from proliferating nuclear technology (e.g. Iran, North Korea, and Iraq), Allison takes the simplistic perspective that unfortunately too often predominates the public discourse. Basically, Allison offers a limited method for resolving international conflicts over nuclear programs. He says that the US and other great powers should offer nuclear fuel at below market price to aspiring nuclear powers with the condition that the receivers do not enrich their own uranium. Allison says that states that reject this economic bargain are obviously up to no good and have nefarious intentions. He concludes that these bad actors should be stopped with military force if needed. I'm certainly not arguing that bribes, sanctions and military strikes have no place in the range of nonproliferation methods; I'm arguing that they are not the only methods. Peaceful and militarized nuclear programs are extremely expensive. States are often not thinking economically when they develop nuclear programs and therefore a simple economic argument is unlikely to persuade them to stop. Allison mentions nothing of national pride, strategic defense, fear of regime, change from the outside or inside, or military-industrial interest groups. Offering security guarantees, starting collaborative science projects, and offering deals that address a state's non-economic interests are all methods that should be included with economic and military methods.
Not to be too hard on Allison. His book is required reading for anyone interested in counter-proliferation. His book is very accessible for non-experts and a (yes) "enjoyable" read. It does an excellent job of representing a popular argument. It is a large and important piece in the puzzle of nonproliferation and conflict resolution, but it is just one piece.
Post-Note: Obviously, I've necessarily simplified the technology and Allison's book in this blog post. However, I hope I have done so while preserving the key concepts.