Just omit the ExpandAll[] call, with other words
Collect[
Simplify[((((((z^2 - qz - 1/27 p^3)/z) /. z -> w^3) /.
w -> 1/6 (3 y + Sqrt[3] Sqrt[4 p + 3 y^2])) /.
p -> (b - a^2/3)) /. q -> ((-2 a^3 + 9 ab - 27 c)/27)) /.
y -> (x + a/3)], x]
in any case you have x in the numerator as well as in the denominator. You can use ExpandDenominator as well as ExpandNominator to control the expansion in detail if needed. You can even include the Simplify in the Collect, then it reads
Collect[((((((z^2 - qz - 1/27 p^3)/z) /. z -> w^3) /.
w -> 1/6 (3 y + Sqrt[3] Sqrt[4 p + 3 y^2])) /.
p -> (b - a^2/3)) /. q -> ((-2 a^3 + 9 ab - 27 c)/27)) /.
y -> (x + a/3), x, Simplify]
