Serre fibration is usually defined as a map $E \to B$ such that for every CW-comlex $Z$ and every maps $Z\to E$ and $Z\times I \to B$ there is a homotopy $Z\times I \to E$ compatible with the others.
Suppose that for the map $E\to B$ this condition is true only when $Z = D^n$. Is this map a Serre fibration?
Serre fibrations are precisely those having homotopy lifting property with respect to $D^n$ for all $n$.
To show this, consider the pushout diagram associated with building your CW complex. You may find the concept of relative homotopy lifting property useful here.