  In this paper we analyse the structure of the Cuntz semigroup of certain
$C(X)$-algebras, for compact spaces of low dimension, that have no
$\mathrm{K}_1$-obstruction in their fibres in a strong sense. The techniques
developed yield computations of the Cuntz semigroup of some surjective
pullbacks of C$^*$-algebras. As a consequence, this allows us to give a
complete description, in terms of semigroup valued lower semicontinuous
functions, of the Cuntz semigroup of $C(X,A)$, where $A$ is a not necessarily
simple C$^*$-algebra of stable rank one and vanishing $\mathrm{K}_1$ for each
closed, two sided ideal. We apply our results to study a variety of examples.
