In an implosion fission bomb, the bulk of the nuclear reactions (100 GJ) occur in the final microsecond--it is a tremendous amount of power--$10^{17}$ Watts. This power is dumped into the compressed Pu pit--which is a few kilogram with a density of 100 g/cm^3-it is small.
The nuclear material is heated to an extremely high temperature, tens of millions of K--it black body radiates (X-rays) to the nearest material: the "cold" bomb components (air quotes because they have just been involved in a conventional explosion). They are heated to a lesser temperature, which blackbody radiates isotropically, heating material further away from the bomb. Note that the atmosphere is opaque to these X-rays, so the process of radiative diffusion continues for several hundred of meters. This is called the fireball--it's all about photons and radiative diffusion. As it diffuses outward, at some point it becomes slower than a shockwave, so that a shockwave separates from the fireball, which is now on the order of a few hundred thousand K. (In photographs of the fireball in the 1st few milliseconds, when it's 100's of meters, the 'shadows' of the bomb components can still be seen).
The shockwave heats air as it passes, but it is not as hot as the fireball. It is at least 50,000K, maybe 100,000K--so it's at least as radiant as a lightning bolt--but it is not transient, and it could subtend a much larger solid angle--hence the phenomenal thermal damage. Nevertheless, it is significantly colder than the material behind it.
As the shockwave propagates, it weakens an eventually no longer heats air to luminance--at this point the traditional fireball has reached it's maximum size. (I say traditional, because its what we see in test footage, but it should be distinguished from the initial fireball which is radiatively diffusing photons with some plasma thrown in--and radiation is the dominant mode of energy transfer, even though the bomb casing can have hypersonic velocities.)
Now for an airburst the shockwave that is reflected from the ground move through heated air, and is thus faster than the direct shockwave: it catches up and the two combine to for a more powerful shock, called the Mach stem. This goes on to produce blast damage, as buoyant forces lift the fireball, producing the infamous mushroom cloud.
For tests like Starfish Prime, which occurred in space--the initial X-rays (very hard X-rays) from the bomb components aren't absorbed by air--they continue to the upper atmosphere where wide scale Compton scattering produces a huge and sudden current, leading to continental scale EMP.
A point of clarification: since the OP asked about shockwave formation--as others pointed out, the temperature rise leads to huge pressure, which leads to a shockwave, but it does not form in the fireball--the radiative diffusion is at first much faster--and it's only as the fireball petters out that the shockwave separates from it.