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I'm interested in the mushroom cloud produced by a nuclear explosion: What is it made of?
I know that the uranium fission reaction is the following (from this web page)
$$\ce{{}^{235}U + {}^1_0n -> {}^{95}Sr + {}^{139}Xe +2~{}^1_0n + 180~MeV}$$
I know that these elements are not stable and decay, so what is left over?
According to The Preparatory Manual of Explosives, the mushroom cloud is produced by the partial vacuum left after initial expansive shock wave. That implies that it is primarily the condensation of water that makes it visible. These clouds form in high explosive as well as in nuclear detonations near the ground. See The Preparatory Manual of Explosives. See also this video.
During the cold war the americans did some work on the chemical form of fallout, the great problem is that shortly after a surface burst or a burst close to the gound the bomb will stir up whatever was close to the detonation or on the ground below it.
The heat of the bomb will cause the dust and other matter to rise up into the firey hot cloud of smoke mixed with fission and activation products together with unused fuel. The nature of the "dust and other matter" depends greatly on the detonation site.
When the detonation site was a coral island then the fallout tended to contain calcium oxide particles from coral plus some iron from the towers which held the bombs. Some of the particles collected were dicalcium ferrite (Ca2Fe2O4). A moderate yeild tower shot in the Nevada test site would give instead glassy spheres. Small yeilding surface shots in the Nevada site gave particles which were partly melted mineral grains.
The chemistry of fallout from underwater explosions was different again, if you want to read more then see C.E. Admans, N.H. Farlow and W.R. Schell, Geochimica et Cosmochimica Acta, 1960, volume 18, pages 42 to 56.
